Metal Gear Solid was my first (console-style) action game and also first stealth game, and it didn't technically "got me" into action/stealth games, because i didn't play any since then, but if/when i had/will have chance (good enough HW), i'll definitely go for Splinter Cell and the likes.
Planescape Torment was my introduction to RPGs, and i tried many others despite being repeatedly dissapointed (yes, even by Baldur's Gate), just because i remember how awesome it was. Regardless, I've never since found an RPG i'd really like, so after every disappointment i install PT again and play it for a while.
(Watching my cousin play) Red Alert 1 got me into RTSes, and the genre was my huge passion for a long time.
(Also watching my cousin play) Civilization 2 got me into 4X+turn-based games, but i've kind of always had trouble finding ones that would really suit me, except all the Civilizations and GalCiv. (I think i may have a problem with too high expectations, as i tend to enter the genre thanks to what is regarded the best game in it, so i'm then usually at least a little disappointed by any other game in the genre i try/play)
Dwarf Fortress (adventure mode) got me into (trying out huge amounts of) rougelikes, and even thinking about making my own one, and again, i was kind of disappointed with most of others i've tried. (Not that they were bad or i didn't enjoy them, but not so much, i just always miss something that would make me feel it is a complete experience).
Doom got me into FPS, and Tomb Raider 1 into action/adventure/platformers.
(Btw, interesting kind of offtopic fact, i never really enjoyed or played too much of Half-Life, i LOVED the intro sequence and the way it was made, but i never felt like i understand/know how to play the game properly, no idea why)
System Shock 2 got me into FPS/RPG hybrids, or generally the kind of games where you spend a lot of time in one level, backtrack trough areas multiple times, but there's always something to do, you feel like you're in a consistent world despite it being divided into relatively small, clearly separated levels, and i'd definitely love to see more of those, despite Thief 1&2 and Deus Ex not really being my cup of tea (Though i love Thief 3... i think the former are a little too hardcore for me? not sure). Yeah, i know this is not officially a genre, but i feel it should be kind of recognized as such, in a way.