@Dale M already gave the punch line, however I believe it is worth going into more details so as to give you material to review the class.
In spirit, your Martial Archetype is close to the Battle Master: the character gains options that they may use at will. This is lucky, because the Battle Master is generally considered to be a good sub-class -- unlike the Champion -- so you should aim to be at about the same power level.
The Battle Master gains 2 class features at level 3, Student of War is a ribbon, leaving the defining feature Combat Superiority:
- You gain your pick of 3 Maneuvers.
- You gain 4d8 Superiority Dice, you regain all Superiority Dice upon completing a Short or Long Rest.
And reading the Maneuvers, you'll see that all (?) require expending a Superiority Dice to use them.
You can see how Stance Change is way better right off the bat:
- 8 different stances.
- Unlimited switches.
- Unlimited use of the bonuses.
And this gives you a hint on how to balance things a bit:
- Limit the number of stances. Should the player take the situational Blind Stance right off the bat, or first pick more generally useful stances? Do they want to be better at fighting, sneaking around, in social situations, etc... they need to make a choice based on their characters' personality, and goals.
- Limit the number of switches/uses, for some at least.
- Small static bonuses such as +2 Damage, +1 (not +2) Attack Rolls, +1 Armor Class, are probably fine, and very range limited Blind Sight is probably fine too (very situational to start with).
- The Neutral Stance is a bit much. Either it should be a small bonus, or limited uses.
- The Counter Stance grants a good damage bonus, but Readying is rarely used since it requires giving up initiative, so that may balance it. I'd still feel more comfortable if it had limited uses.
- The Eagle Stance is too much because it comes on top of the Archery Fighting Style, granting +4 to Attack Rolls when combined. Limited uses would solve the issue.
- The Wind Stance is overpowered. Push back is very powerful, even limited to 5 ft., one push back a turn, at no cost, using the best DC the character can put forward for their level, will allow them to bull their way through anything. All Day Long. Limited uses is definitely warranted.
- Allow scaling. Just like the Battle Master, you can allow the player to pick more stances as they gain levels, and gain more uses of their limited use abilities.
I would also note how the Battle Master bonuses scale with levels, because +2 Damage at level 3 is fairly impressive, but at level 20 it's quite underwhelming.
Then, with regard to other levels, the Battle Master is entirely built around Combat Superiority.
The only features they gain beyond that are:
- At level 3, a ribbon: Proficiency in one Artisan's Tools of their choice.
- At level 7, another ribbon: 1 minute of observation to learn the relative ranking of enemy compared to themselves.
That's it.
By comparison, the 7th level and 10th level abilities of your Ronin are much more powerful:
- Without A Trace: you could make it a ribbon, downgrading it to just gaining Proficiency in Stealth. Remember the Ronin would gain some more stances/more uses of stances at level 7 on top of that.
- Calm Thoughts: much better than Know Your Enemy.
- Saving Throws are divided in 2 groups, the High-Frequency Saving Throws are Dexterity, Constitution, and Wisdom, the Low-Frequency Saving Throws are Strength, Intelligence, and Charisma. You'll note that by default each class grants 1 High and 1 Low, and this is not a coincidence. As a result, you should never give a character another High-Frequency; if they pick the feat Resilient (Dexterity), they'd cover all of the most frequent saves, and you'll find yourself in trouble when it comes to challenging them.
- And on top you give them Advantage on 2 frequent uses of the Wisdom Saving Throws.
- All of that on a Fighter which has Indomitable allowing them to reroll a Saving Throw thrice a day.
- Advantage + Proficiency + Reroll: they're unstoppable, or really unlucky.
- "Advantage vs Frightened" alone would be better than Know Your Enemy. I'd advise downgrading to at least that.
I would also note that instead of venturing in the dark -- it's really hard to write a balanced sub-class from scratch -- you could stick much closer to the Battle Master:
- Replace Student of War with Without A Trace, only granting Proficiency to Stealth.
- Replace Know Your Enemy with Calm Thoughts, giving Advantage vs Charmed.
- Cherry-pick some of the Maneuvers, and replace the others.
- Don't forget to check Tasha's Cauldron of Everything, p. 42 for more Maneuvers.
It's much easier to write a Maneuver. Most of them are:
- Either "Expend 1 Superiority Die, add result to X"
- Or "Expend 1 Superiority Die, gain Advantage on next X"
Follow that template, and X can be pretty much anything without causing balance issues.
And of course, you may want to double check the Samurai archetype from Xanathos' Guide to Everything, p. 31. They are one of the few sub-classes getting an extra High-Frequency Saving Throw; at the cost of getting less of everything else.
Depending on your group, it may not matter if it goes up to level 20 as a lot of groups campaigns dissolve before that point especially with new players.
Along with this, this class is at least partially inspired by Matthew Mercer's Gunslinger which isn't official, but normally widely respected (though I can't say for this class specifically). As a result you may be better using that which is a archetype of fighter, likely very balanced, and fully formed.
Before going through the entire thing, a couple of questions/notes that may effect the rest of your campaign:
ASI:
There's normally an optional rule to swap out ASI for feats if the player wants to, is that allowed?
Ammunition:
Some groups tend to, at least partially, ignore how much ammo people with bows are actually carrying. Are you going to do this with your group? I don't believe gunpowder has a rulebook suggested cost so you'll have to figure that out, maybe just using arrow costs.
Weapons:
Your weapon list is likely referring to homebrew weapons, whilst some guns exist in the PHB the revolver does 2d8 which is overpowered for a normal starting weapon at range. I suggest reskinning a crossbow or something similar for all these suggested weapons. I have no idea what the grenades are meant to be in context.
Overall the class isn't really in a reviewable state, some things referred to simply don't exist in the linked content, such as Skill Shots which are part of the gunslinger class linked above (though called "Trick Shots").
Here are a few notes if you want to continue your version however:
Wary:
Slightly weaker version of the alert feat, this is campaign dependent but will likely have a large effect on the campaign if it has any effect as it will in essence make the party un-surpriseable.
Grit/Skill Shot:
To much for level 1, this is an archetype ability (and also named differently in 2 places), this should be given at level 3, if you think the game would be to boring starting without such an ability simply start your campaign with characters at level 3.
Shooting Styles:
This is a slightly less variable version of Fighting Styles, take a peak at either Ranger who also get this at 2nd level if you want to give more options (Though places may list UA content too, which you probably shouldn't allow).
Gunsmithing:
Level 3 is usually your archetype which is a lot stronger than making a weapon, which may be a slow process and awkward to balance for a new DM. As stated before, this should probably be Skill Shots.
Extra Attack:
This is a holdover from its Fighter origins, make it 5th, 11th, and 20th.
Bullet Time:
This is a strong ability to give. Doing an average 48 damagebefore crits and if you use a reskinned crossbow as a your gun (This class will have 20 Dex by level 10). Classes normally get a semi decent thing for level 10, take champions for example:
Additional Fighting Style
At 10th level, you can choose a second option from the Fighting Style class feature.
Which can be reskinned for Shooting Styles.
That being said, a fighter can attack 4 times once per day at this point 1 level later they can do 6, and at level 20 they can do 8. So this would just be a worst fighter at high levels.
Best Answer
A good start, but there are issues
This subclass is a pretty decent start for a new homebrew. The idea and thematics are solid. However, some of the mechanics need a bit of work. I'm going to mostly compare this to the Eldritch Knight and the unofficial Gunslinger fighter subclasses, as they the closest existing content.
Firearm Rules and Proficiency
It seems you are using the RAW firearms rules. That's fine, but most people find them overpowered and a strict improvement over standard weapons. Matt Mercer's Gunslinger subclass provides a pretty solid alternative ruleset for them. You would need to provide proficiency as well, but that's not a major issue.
Broken Spell Progression
There are multiple issues with the spell progression you have implemented. Unfortunately, all of them are overpowered.
I strongly suggest you change your progression to match that of the Eldritch Knight. If you do, it will solve all of the listed issues above. It will give you (1) a progression of Spells Known, (2) correct spells per level and also (3) grants cantrips. Overall, adopting the progression of other casting subclasses is the best way to make sure your subclass fits the game.
Arcane Bullet
This is the base feature of your entire subclass and is understandably a big part of it. However, there are still problems with it.
The arcane bullet feature needs a lot of work. What you are trying to do is difficult to balance into the 5e design philosophy. However, I suggest you look at the Gunslinger and Arcane Archer subclasses and draw some inspiration from there. (Self plug alert) You might also like to look at my Enforcer homebrew class for an alternative take on what you're doing here.
Ricochet
Cool feature, but you need to limit the number of uses. Proficiency or intelligence modifier per long rest seems reasonable.
Adaptable Shot
Reasonable feature for the level, but combined it with Ricochet and you are doing a lot more at 7th level compared to other Fighter subclasses. (Also, I don't understand how Adaptable = Magic? I would rename this feature.) Additionally, most fighters will have a magic weapon by 7th level, making this largely pointless.
Elemental Shot
Why make this restricted to a rest to change it? If you want it to be difficult to change it in combat, make it take 1 minute. Otherwise, just let them decide when firing. It's not overpowered, since Way of the Ascendant Dragon monk can do this from 3rd level. My suggestion is to combine this and Adaptable Shot into a single feature at 10th level.
Snap Shot
Another nice feature that needs some limitations on number of uses. Again, Proficiency or intelligence modifier per long rest is a good place to start.
Caster Shell
One casting of a higher level spell per day as an 18th level feature seems reasonable to me. It's not quite in line with what other Fighter subclasses get, but its not really broken either.
Conclusion
Overall, this class is extremely overpowered, but mostly due to problems with spell progression and the specific mechanics around Arcane Bullet. Fix up those things and you have a decent start for a homebrew subclass.