A draconic fighter is not something I can remember seeing before, so I'm curious how you're going to go about this.
Mechanics
Dragonstone Transformation -
+1 to AC is good, but needing to use light or Medium armor means this just puts you on equal footing with heavy armor. The rest of these abilities are cool, but feel underpowered. The weapon is a little bit better than a whip and Arcane Blast is comparable to a Dragonborn's breath weapon (which is generally seen as underpowered). Arcane Blast is probably fine, but would you consider bumping the damage die of the arcane spear to 1d8?
Edit: I noticed I'm directly contradicting an earlier comment from nickl_2000, so I'm going to math this out and compare to the Battle Master.
Spoiler: Math
Show
At 3rd level, a Battle Master using a whip can do an extra 4d8 damage with it, average of 18. With the arcane spear doing 1d6 (and an average of 1 more damage per hit than a whip), it would take 18 attacks to draw even.
At 11th level, the Battle Master does 5d10, for an average of 27.5 damage. The arcane spear will take an equal number of attacks to draw even and even at three attacks a turn, it still takes 9 rounds.
At 20th level, the Battle Master does 6d12, for an average of 39 damage. The arcane spear at four attacks a turn will take 10 rounds to draw even.
With the damage being 1d8, 3rd level is closer (9 attacks), 11th level is a reasonable 4-5 rounds, and 20th level is 5 rounds.
I'm standing by my statement: It could be bumped up to 1d8.
Is there a reason the transformation has a specific number of uses instead of being based off an ability modifier?
This feature does an excellent job of showing your draconic nature, which is really cool.
Blood of the First Dragons -
That's an uncommon resistance and it continues your established theme. Balanced and good synergy.
Dragonfang -
Oh, that's cool. The math of Arcane Blast in place of an attack is quite on par with a Battle Master using all of their Superiority Dice to add to damage. 4d6 to ~2 creatures versus 5d10 total damage to single creatures. It's even a little weaker because of the saves.
Spamming Arcane Blast at this level is no issue and I like that the option now exists to do so.
Dragon-kin -
Woohoo, flying Fighter!
Overall
Everything works together well and builds atop what has come before. I'm looking forward to seeing what else you can do with subclasses in future competitions.