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.

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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.

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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.

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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).

Verdict:

This race is too strong, and includes one ability that, even on its own, is too much to grant to a race.

Explanation:

I'm making a breakdown using this guide for balancing new 5e races. It suggests that on average, a race is about 6 points worth of options. Most of the suggested abilities for your race are listed there, and counting them up, you end up with around 9 points. That means they're on the "very strong side" of races (beating out the Mountain Dwarf, which is currently the most powerful race), and could do with some fewer options. Especially considering I am not counting extra points for flexibility with the skills and +1 ability point.

However, that's ignoring one thing in your race which is incredibly powerful, way beyond anything another race gets (and pretty much forces the race down a single path) and that's the ability to cast the innate spells at your highest spell slot level.

This means that your race is able to cast 3 extra spells at their highest level spell slot every short rest, which is incredibly powerful. No other race has a leveled spell that's available every short rest. No class besides the Warlock even has that, and the Warlock only reaches 3 spell slots per short rest when they hit level 11.

Additionally, as they grow in level, they will ultimately be able to cast 4 9th spells per day, while any other character in the game only gets 1. (And the same applies at lower levels). Since all 3 spells you picked for them to get benefit from being cast at higher level, this is incredibly powerful; way beyond anything what other races grant. It also pretty much forces you down a spell-casting path, as you'll be missing out arguably the most powerful racial trait in the game.

Also, I'm not counting any point deduction for the vulnerability to Radiant damage. The number of creatures in the book that deal Radiant damage is next to none, so that trait isn't to come up very often.

Full breakdown of points

• Ability scores: 3 points (not increasing it for flexibility)
• Darkvision: 0.5 points
• Necrolyte magic: 1.5 points (ignoring the part about the spell slots scaling)
• Necrolyte training: 1 point
• Necrotic skills: 1 point (ignoring the flexibility, again)
• Fetid Attack: 1 point
• Radiant Weakness: I'm ignoring this because it comes up almost never and races usually don't have Weaknesses
• Languages: 1 point for 2 extra languages
• Everything else: 0 points; those are baseline

Suggestions

Definitely drop the scaling spell slots on the spell-casting. I don't think you'll be able to balance that. You should cut down on the superfluous features a bit; I'd suggest dropping the bonus skills, the unarmed attack and maybe one of the languages and/or the Darkvision. Dropping all of those would bring you to 6 points, which is balanced with other races.

Maybe even replace the two languages with "one language of choice". That still gives them a bit of that human flexibility (along with the +1 ability) that comes being based on them, while not making them too strong compared to other races.

I'd also drop the Radiant vulnerability. Not even most genuine Undead have that; it's unlikely to come up, and the one time during a campaign it comes up, it might well end up killing your character (or just being forgotten because it's never come up before.)