Survival Horror - Fatal Frame. I picked it up at Blockbuster once after I had played all the games that looked good; then I about peed myself. I had played a few survival horror games, but I never got very far into them before getting freaked out and/or giving up (SH2 confused the hell out of my 12-year-old brain "WHERE ARE THE LEVELS??!! I'm lost!! This game sux." *walks off shakily*). Around the same time, I picked up Resident Evil 4. So, I actually had a couple of games that really launched me into seeking out other horror titles. I went back and played all of the old Resident Evil games (except the 1st one; I played the Gamecube Remake instead). After that I branched out into Silent Hill, Clock Tower 3, eventually Half Life 1, and so-on.
Point-and-Click: I have a hard time separating which was 1st and which was 2nd so: King's Quest VII and/or Mr. Potato Head Saves Veggie Valley...Shut up that game was AWESOME.
FPS: Doom
Beat 'em Up: The Mighty Morphin Power Rangers: The Movie (SNES)
Open World RPG: Morrowind
JRPG: The first one I can really remember playing (watched by brother play a lot when I was little) was Suikoden 1. I'm not into JRPGs much anymore, but between the ages of about 10 - 16, this was one of my favorite genres largely due to how much fun that game was.
Platformers/Adventure?: The Adventures of Dizzy. I recently found a copy of this to play again. It's still ridiculously hard. It was one of the first I played (Mario obviously being THE first), but it was really interesting because it seemed much more complicated and focused on semi-free-roam level design than a typical platformer. I may also just be categorizing the game wrong.
Stealth: Splinter Cell