I'll dissect this homebrew race on a per-trait basis, and then offer an overall evaluation at the end:
Neutral traits
These are the traits I find are neither over or underpowered:
Ability Score Increases. Your Strength score increases by 2, and your Dexterity score increases by 1.
Pretty standard, nothing under or overpowered here.
Age. Lycanthropes mature at the same rate as Humans, and live up to around 150 years.
Alignment. A Lycanthrope is typically of a Chaotic Alignment.
Size. Lycanthropes stand 6 to 7 feet tall just as a Human. Your size is Medium.
All fluff.
Speed. Your base walking speed is 30 feet in your humanoid and hybrid forms, and 50 feet in wolf form.
This is better than what most races get, but it's situational enough to not be anything of note.
Darkvision. You can see in dim light within 60 feet as if it were bright light, and in darkness as if it were dim light. You can't discern color in darkness, only shades of gray and red.
Pretty standard. I like the addition of the red color.
Tracking. You can detect opponents within 15 feet by sense of smell. If the opponent is upwind, the range increases to 40 feet; if downwind, it drops to 10 feet. Strong scents can be detected at twice the ranges noted above. Overpowering scents can be detected at triple normal range. When a creature detects a scent, the exact location of the source is not revealed—only its presence somewhere within range.
This is a cool trait, but I'm doubtful as to its actual usefulness in-game. The range is so short that you're still going to get surprised, and tracking whether you're upwind or if the creature is smelly sounds like it will quickly get tedious. Consider giving the Werewolf Keen Smell or Keen Senses (Proficiency in Perception), like an Elf.
Shapeshifter. You can take 1 action to change into your Hybrid or Wolf Form. While in Hybrid Form you can not wield weapons, or wear armor. While in wolf form you cannot wear armour or wield weapons. You are able to wear certain equipment at the DM's discretion, such as circlets and robes. You can choose to have equipment you are wearing to either merge into your form or be dropped to the ground. You gain no benefit from equipment that is merged into your form.
The ability to change, itself, is pretty standard, however, the restriction on weapons and armor are weird (They can wield Foci and Holy Symbols, but not weapons?) This trait is probably the biggest counter-balance to all the positive traits because of the limitation on what you can hold in your hand.
Languages. You know Common, Wolf and one extra language. You can only speak in Human and Hybrid forms.
Pretty Standard, though I wonder why you can't speak Wolf in Wolf form.
Wolf Empathy. You can communicate with wolves and dire wolves.
Exceptional but not overpowered. Gnomes can speak with small animals, which I've seen can be useful at times, this trait is going to be a lot more useful than its gnomish counterpart.
Positive Traits
These are the traits that your Werewolf race gains as benefits, which I rate as over or underpowered based on comparison with existing traits from other published races:
Regeneration. You have resistance to damage from nonmagical weapons that aren't silvered. At the start of each of your turns, you regain hit points equal to 1d4 + your Constitution modifier if you have no more than half your hit points left, and have at least 1 hit point.
Overpowered. This trait effectively gives the Werewolf race 1.5x to double HP, with the resistance alone. Similar races with resistance such as the Dragonborn (1 element), Dwarf (poison) and the Aasimar (radiant, necrotic), grant only 1 or 2 types of resistances and only for situational damage types- a resistance to the three most common damage types is too strong.
The actual regeneration trait is also very strong. The only race I found with regeneration is the UA-Gothic Heroes Revenant subrace, and the Revenant only regenerates 1 HP per turn, your homebrew has the potential to regenerate 5 HP, with a +3 Con, at 1st-level.
Natural Weapons. While in your hybrid or wolf forms, you have a 1D6 Bite and 1D8 Claw attack. Increase damage at level 5 to Bite 1D8 and Claw 1D10. On a successful melee attack you may use your bonus action to make a claw attack on the same target.
This is tricksy. The d6 Bite is ok, but the claw attack basically grants you access to a non-light one-hand Martial weapon you can use to attack with a Bonus Action, normally you'd need Dual Wielder to pull that off. Taken in conjunction with the Shapeshifter trait where you can't wield anything in your hands when you transform, it sounds ok, but then there's gonna be the powergamer who will want to play a Monk in order to get the early d10 unarmed strikes, with flurry of blows.
Then again, if this race will be paired with any class that isn't a monk, he's gonna have no access to magical weaponry, and be forever at the mercy of creatures with resistance to Bludgeoning, Piercing and Slashing damage.
Trip Attack. While in Wolf or Hybrid form you can attempt to trip the opponent as a bonus action after a bite attack. The opponent must make a Strength saving throw equal to 10+ your Strength modifier + your proficiency bonus.
Flavorful and only slightly overpowered- most save DCs are computed as 8 + modifier + proficiency.
Skills. You are Proficient in Survival, Nature, Intimidation and Perception rolls. You have disadvantage on Persuasion checks.
Overpowered. A Half Elf with Skill Versatility only gets 2 skills, this trait grants four. The Disadvantage with Persuasion doesn't counter balance this enough (nor does it make much sense, do you still get Disadvantage in Human Form?).
Saving Throws. You are proficient with Strength and Dexterity saving throws.
Overpowered. No race grants Proficiency in saving throws. These are strictly benefits from picking a class.
Tough Hide. While you are not wearing any Armour or shields your AC becomes 10+ Strength Modifier + Proficiency Bonus (All Forms)
Overpowered. Unarmored Defenses usually don't include Proficiency Bonus as a component. This is because Proficiency is "free" as the character levels up, and it will quickly get out of hand at higher levels. Consider that at 17th-level, and at +5 strength, this character will have 22 AC, even before items that boost strength like a Belt of Giant Strength.
Negative Traits
These traits would be the counter-balancing factors for the Werewolf race. I will say, in advance, that these traits are not effective counter-balancers (it's not as discouraging as, say, Sunlight Sensitivity) for all the Positive Traits above. I'll explain why below each one.
Silver Weakness. You are vulnerable to damage from silvered weapons.
Vulnerability is normally really punishing, but in this very specific case, when the vulnerability comes into play, it will feel like you're specifically targeting the PC, As Erik puts so well in another answer.
Also consider that not even actual Lycanthropes in the MM are vulnerable to silvered weapons, and no official race gives you vulnerability to anything.
Full Moon Rage. At the start of a night with a Full Moon you must make a DC13 Wisdom save or lose control of yourself. During this time the DM has Control over what your character does, however every hour you may reroll your saving throw to take back control. Once you make the correct saving throw you retain control for the rest of the night.
Not a huge counter-balance, it will only come up once a month, and the DC is set pretty low (with a chance to break it every hour). What's more, smart players will make preparations if they know the full moon is coming up. They're going to tie themselves to trees or manacle themselves to a cell, which negates what this is supposed to accomplish.
New Moon. During a new moon, you cannot transform, and are damaged normally (without resistances and weaknesses).
As above, it's not a huge counter-balance. Smart players will avoid adventuring at this time of weakness. If you force them to adventure in this scenario, it will feel antagonistic.
Overall, this race is overpowered, almost broken. What's really glaringly overpowered are the Regeneration, Saving Throws, Tough Hide, Skills, and how the Natural Weapons scale so fast. The Werewolf race will outshine just about anything at low-to-mid-levels, and be powerless against higher-level opponents with resistances.
The race has too many things going for it, if you take a look at other races, they usually have 3 or 4 defining traits (not including ability score bonuses, darkvision, languages, etc), this race has a whopping 8.
Using this race power-level analyzer, from reddit's /u/aranim and /u/JamesMusicus, this Werewolf scores:
- Ability Score Increase +3
- Silver Weakness -1
- Regeneration +2 (from regeneration) +1.5 (from resistances) = +3.5
- Natural Weapons +1
- Tracking +0.5
- Darkvision +0.5
- Shapeshifter +0
- Trip Attack +0.5
- Wolf Empathy +0.5
- Skills +2
- Saving Throws +2
- Full Moon/New Moon weakness -2
- Tough Hide +2 (being conservative)
- Language +0.5
Total 13 points, nearly triple the score of the poor Dragonborn (4.5), and eclipsing the highest-rated Mountain Dwarf (8).
Okay, this is a lot. In order to actually be able to answer it without writing an essay, I won't make complete comparisons for each subrace. As I mentioned in a comment, if you want a detailed feedback, I would create a separate question for each subrace - they are too different from each other.
Your Base Race has almost no features, which is fine considering the main thing is on the subrace. It's still worth to note that consistently the base race is the one giving a +2 AS bonus.
Earth
You get a +3 AS, which is the usual for playable races. +2 Str +1 Cha is an awkward distribution though (same as dragonborn), probably most useful for Paladins (usually the class Dragonborns are played). For most of other classes, one of the two stats will be kinda wasted. By itself, this doesn't make the race weaker. Tiefling gets an even worse bonus distribution.
Barkskin is a 2nd level spell. Not only you are giving a free 2nd level spell to the character, you are giving a free 2nd level spell slot as well. We could enter the discussion if barkskin is a strong spell or not (my opinion: it's not, actually it's weak), but the fact is: 2nd level spells to 1st level players is alot.
I would change it to a feature that might scale with PC's level, similar to Breath Weapon from Dragonborn.
How strong will your resistance to piercing be depends on the campaign. While Dragonborn and Tieflings have stronger resistances (elemental), they are more useful in later stages of the game, while piercing might be too strong in early stages, where physical damage is most common.
I would say the Earth subrace is (maybe too) strong in Tier 1 adventures, but becomes balanced from Tier 2 onwards. I'm not sure about Tier 3, but it becomes kinda weak in Tier 4, compared to the playable races.. One easy way to fix it, as I mentioned, is making the Barkskin feature scale with the tiers. I will make a note here that is valid to the following feedbacks, though: The bonus from races usually are less important in the endgame anyway, since your class features and magic items will compensate for it.
Air
Similar to Aarakocra. I am not sure the +10 move speed in land and Hover compensate both the -1 Dex and -20 Move speed flying, though. Also, I'm not sure how mechanically useful the weather prediction thing will be, that depends on you. Also, while Aarakocra +2 Dex +1 Wis helps alot with Rangers and Druids, the +1 Cha instead makes it a little more awkward, probably being more useful for Bards. Sorcerers might be interested in the flying feature, though, and get more from +1 Cha than +1 Wis.
As it is, it's clearly weaker than Aarakocra. (Yes, I know you asked "compared to PHB", but since PHB doesn't have flying races, I'm forced to compare it to EE, sorry). That said, it isn't completely underpowered since the AS bonus is more attractive for bards and sorcerers.
Fire
My first impression was god it is too strong. Then I noticed that, while you added a lot of powerful features together, they don't have too much synergy.
The main problem is that, from internal comparison, unless Flying can compensate it very well, it's way stronger than the Air version.
Other than that, it's very similar to a Red, Brass or Gold Dragonborn, changing the Str for Dex.
In my opinion, having the Fire Bolt is weaker than the Breath Weapon, even if it can be cast as many times as you want, because you probably will have other ways to deal damage anyway. Thinking in Dex based characters, they will probably be shooting arrows instead of fire bolts. The Breath Weapon is AoE, though, and usually cast by a class that doesn't have that much AoE early (Paladin), so it's more useful in the specific situation it's actually used.
The Grappling damage would also be more useful if it was a +2 Str bonus instead. Building a grappling character without a Str bonus is too suboptimal. I can't even guess how strong this feature will be as it is right now, sorry but you'll have to playtest.
So, comparing to Dragonborns, it will depend on how much Darkvision is a think in your setting. If darkvision is really important, it's balanced compared to Dragonborn.
If we compare it to Tieflings, it seems to get outscaled. Early on, you have a damage cantrip and a grappling feature, while Tiefling has a flavor cantrip. From level 3 onwards they get spells, though, while the scaling of this subrace is only based on the fire bolt cantrip. Again, even if it gets outscaled, it's not an obvious choice since your AS bonus is more attractive to Dex users.
Still, I would fix the fact that the AS bonus (dex) and the features you are giving don't synergize well.
One possible solution is to change the Fire subrace to +2 Str and the Earth to +2 Con instead (if you don't want both to have the same AS bonus). It would fit the "durable" flavor of the Earth/stone and the "aggressiveness" of the Fire.
Note that, if you do, the Fire subrace becomes a very strong option for grappling buildings. I wouldn't say this makes it overpowered, though, since grappling builds are usually more for fun than optimal.
Water
This one is hard to compare. The Healing feature seems weak (just Your Level is not much HP. Compare with Dwarves getting +1 HP/level, which is arguably stronger). Create/Destroy water is a 1st level spell, but this one doesn't need much discussion about being strong or not - it's not strong mechanically.
So it's about the other 3 features. First, the AS bonus: +1 Con +1 Cha is weak. Variant humans get to choose their two +1s and get a Feat on top of that. But they are arguably the best race in the game, so let's not do this. Still, +2 AS is underwhelming and I would expect the features to compensate it strongly. For the Air case, flying might do it, let's see the Water.
Water Breather again will obviously depend on your setting. It's arguably harder to use than Darkvision or Flying, though, since while your party can easily fight in a dark cave or an open field, fighting under water is way more unlikely and usually not desired by PCs.
That leaves Armor of Agathys to compensate the underwhelming AS bonus. Well, sadly it doesn't. It is a very, very strong feature in the first two levels, where +5 HP and 5-10 damage to the enemy are alot, but it quickly becomes weak as you progress in the game. Again, I feel you are lacking scaling. Increase these numbers as the tiers go up. So, similar to the Earth subrace, I'd say it is (maybe too) strong in Tier 1, but gets weaker (and even underpowered) in Tier 3+.
TL;DR
My main criticism here is that all your subraces lack scaling. While Air is fine without it, since Flying itself has a constant utility through the game, the other subraces might suffer from it. It is fine if the campaign is focused in Tier 2, but they seem too strong for Tier 1 and too weak for Tier 3+. Again, note that being too weak in Tier 3+ is not that important, since you have lots of class features and magic items to compensate for that. Still it's something you might want to take into account.
I suggest scaling as a way to fix it, but introducing mechanics that have utility through the game (such as having advantage on something, proficiency on something, etc) also works.
It's worth to mention, though, if you are playing a long campaign that goes through, for example, levels 1 to 15, your subraces are mostly fine, as they will be strong early on and weak later, and it's a valid trade-off for your players to choose. I have linked a chat discussion about this, if you are interested.
Side note:
If you are wondering why I'm not comparing to Dwarves, Elves, Gnomes or Halflings, which also don't have numbers scaling with the levels, it's because they have proficiencies in skills, weapons or tools, or advantages in saving throws, which are the things scaling. Also, they get lots of other features, such as Trance, Luck, etc. Comparing to Dragonborns, Tieflings and Aarakocras is easier and more fair (imo) because they are closer to what OP has made. It's hard to compare if proficiency in perception, using daggers and sleeping 2 hours less is better than 1d8 damage when grappling.
Best Answer
Pack tactics is probably too strong, wolf bite doesn't scale and change form is too weak.
Overall, this jibes fairly well with what you're looking for when you are creating a custom race, though it's probably not ideal and I'll explain why in a minute. First let's address the specifics:
Pack Tactics is typically a monster quality, it's probably not super OP for a PC, but seems a bit stronger than I'd want to grant as a racial power. However, this is also basically an implementation of the optional flanking rules/a slightly stronger version of the rogue class power (which is honestly the bit that has me worried, it's a better version of the feature that lets the rogue use SA).
Wolf Bite. Consider making this 2+proficiency rather than +4, as a non-scaling attack will quickly become useless. 2+ prof makes it slightly worse to-hit than a good melee attack, but the upped damage should account for that nicely.
Change form. This is the feature I have the most issue with. You basically make the wolf form amazingly easy to kill, and thus pretty easy to incapacitate the character, you also make it hard to access (a full action) and penalize it heavily with disadvantage for a round. I'd definitely reconsider this. You don't want them to be a full on druid, but it seems like they should be combat effective as a wolf (even if the primary purpose of the form is not combat).
I have a couple of other concerns, I'd probably grant proficiency rather than advantage on the transmutation spells (this is hair splitting though, they end up about equal by the end, but advantage is much much better early).
The ultimate question to ask yourself when creating this is the same one that the DM suggests asking when you create a feat or other game element "Is this feature so powerful that everyone will want it" if the answer is "no" then you've got something you can work with in your game. Personally, looking at this I don't see anything about this race that would make it signficantly favorable to every class in the game. It strongly favors some melee characters, but there isn't enough here to make it a "must take" for even any one class (though it might be a bit close for the rogue).
And that last bit "in your game" is the important factor. Judge your homebrew on your own game. How optimized are characters typically? How likely are your players to exploit game breaking combos? How carefully you have to design will depend greatly on these things.
The one place this does sort of fall afoul of the DMG's advice is that it ranges a bit from stealing things from other races. This makes sense though as you're building a were template and some of the things you need to have just aren't present in the existing races (though they are in the were-templates in the monster manual, you may want to spend some time reading those for ideas on how to implement the were-aspect of your race, I'll fill in some suggestions when I get in front of my book).
The only thing I feel is missing here, and it may be a setting/conception issue more than a real problem is the lack of a hybrid form. All of the were-templates in the MM have one and it might be helpful to be able to sit between the world of wolf and human for a bit.