It's hard to talk about when I first became a gamer because I always have been one. I can't remember what my first game was because I was playing games while I was still in diapers and before I could talk. Video games have always been an important part of my life-- there was no "start" to it that I can remember!

Edit: If we are talking about genres in specific, then I have to credit Starcraft for really introducing me to strategy. I played Sim games and other semi-strategy-ish games before that, but Starcraft back in 1999 (or whenever it was released) really opened up a whole new world for me, and because of that, it will always have a special place in my heart.