Fighting: World Heroes - I was pretty much a fan of the fighting genre the first time I laid eyes on them. World Heroes is just the first one I managed to get my hands on. Been playing pretty much any fighting game I can try ever since
Racing: Bill Elliot's Nascar Challenge - Again, I was already attracted to racing games from the get go. I'm pretty sure I played older racing games on the PC, but this is the first one that I actually remember. I don't think I was really a fan until Need for Speed Underground though.
Western RPG: Drakkhen - My mom got me this game when I was a kid, and I really only chose it because I liked the cover. I actually finished it, but it must not have been very good, because I don't recall touching another Western RPG for a few years until KOTOR if I remember correctly. I don't think I really became a fan of western RPG's until Fallout 3 and Mass Effect though.
JRPG: Chrono Trigger - Probably one of the first games I chose for myself and not because someone was buying it for me or because I had seen a friend playing it. Definitely made me a fan of the genre, so much so that I still prefer the old sprites to the super pretty JRPG's of today.
FPS: Doom - One of my friends had it on his computer as a kid (and another one called "Blood"), and I'd play it whenever I was over at his house. I wouldn't say it made me a fan though. I'd say Goldeneye took that title.
Third Person Shooter: Winback - I recall enjoying the game quite a bit, but I wouldn't say it made me a fan of the genre since I don't remember playing many third person shooters until the current generation, unless you want to count Resident Evil 4 as a third person shooter.
Sidescrolling: Teenage Mutant Ninja Turtles - This was actually also the first console game I ever played, again, at a friends house. It made me a fan, but then, I was a kid and this was Ninja Turtles, so it didn't take much back then.
Horror: F.E.A.R. - I suppose this one's a little subjective, since I played Resident Evil before this, but since this is the first game that actually scared me, I have to say that this is the first horror game I ever played, and definitely the one that made me a fan of the genre, actually prompting me to backtrack to older games like Silent Hill and Fatal Frame.
Shooter: Top Gun - Anybody who watches Angry Video Game Nerd or has played this knows that it's damn near impossible to deal with, so while I liked the game, it didn't make me a fan. However, like fighting and racing games, as a kid, I was naturally drawn to shooters, or anything with planes in general. Star Fox made me a fan of the shooting genre, with an honorable mention to Lethal Enforcers for being the game that made me love Light Gun Shooters as a kid.
And that's all I remember..... which is good, since this post is too long as it is.