First platformer: not sure which was the first, but Henry's House was one of my favourites in my early gaming career. Even before I got hold of a machine of my own. Played it a lot on a borrowed C64.
First racing game: Outrun on the C64, maybe. Again not sure. It's been so long now, and details are fading fast.
First adventure game: This one I'm pretty sure of - Murder on the Mississippi (C64). Great sherlock holmes style detective game. Didn't get very far though, and I kept dying and had no idea how to save.
First shoot'em up: even more unclear. Airwolf (C64) perhaps.
First arcade/misc: Space Taxi (C64), most likely not my first, but certainly one of my favourites.
First coin-op: Bubble Bobble (C64). Played it a LOT, one of my favourites of all time even today.
First fighting game: Yie Ar Kung Fu (C64). I still enjoy this. I also like the the clone by Team 17, called Full Contact.
First soccer game: that's easy - International Soccer (the one by Commodore on C64). I remember playing it to death, and then some. I used the cartridge version.
First point'n'click: Zak McKracken (C64). Playing it with a joystick wasn't as good as when I got it on the Amiga, but I absolutely loved the game.
First strategy game: Defender of the Crown (C64)
First ever completed game: I'm 99% sure that The Secret of Monkey Island is the first game I ever completed.
First ever RPG: Bard's Tale or Times of Lore on the C64.
First ever JRPG: Final Fantasy VII. Still among my favourite games ever.
First ever stealth game: Metal Gear Solid I think. Not sure if I played Tenchu first, or if playing MGS sparked my interest in Tenchu when I heard what kind of game it was.
First ever FPS: Doom or Rise of the Triad. Though I didn't have my own PC - I was still hoping for an Amiga comeback.