let's see, I'll just put up the ones I can remember I suppose a lot of these games will be for the genesis / mega drive as it was my first gaming system.
First fighting game: Golden Axe I (Gensis / mega drive)
First Platform game: Disney's Aladdin (Genesis / mega drive)
First 3D platform game: Some platform game for PS One, I don't know actually...
First first person shooter: Duke Nukem 3D (computer)
First RPG: Fallout (Computer) might be Diablo (computer) as well, I'm not sure.
First Third person shooter: Duke Nukem 1 (computer) if that counts as a 3rd person shooter, else I don't know. Quackshot maybe (genesis / mega drive)...
First adventure game: The secrete of monkey island (computer) or Indiana Jones and the Last Crusade (computer) I honestly don't recall, it's one of them, I got ahold of them pretty much at the same time
First Japanese RPG: Final Fantasy 7 (Playstation One)
First game I beat: uuuh.... that game with the natzi zombies I think, don't recall the name, you're in a pyramid? maybe?
First horror game: Mummy: Tomb of the Pharaoh (I recall really liking the recorded movies and the environment scared me when I played it first time.. and the second time too.... thinking on it, I always played it with my brother because both of us were to scared to play it alone

children for you)
First sports game: Some game for the super nintendo console, I think it was ice hockey
First racing game: Some game for the super nintendo console, I recall always getting eaten by the train at my friends place...
First online game: Myth II: Soulblighter (computer)
First RTS: Command & Conquer (computer)
Well, those are all I can think of, enjoy