I don't even remember. My formative years were in the mid-to-late 90s, so a lot of the first games I played on PC were educational games from the DK company (Dorling Kindersley publishing). I remember one game about this golfer on an island... but I don't remember what the goal was.
My first console game was on the NES; Super Mario Bros.! I played that game through and through so many times that the music is permanently embedded in my head, and I remember the layout of most of the levels (my favorite ones are in Worlds 3 and 6 because they're at NIGHT!). Tetris and Rad Racer were my second choices, while my father preferred Super Sprint. Yesiree, a lot of memories on the ol' NES.
The next console we obtained was a GameCube back in 2008, followed by a Wii in 2009 or '10. On the GameCube the first games I played were Need for Speed: Hot Pursuit 2 and Animal Crossing. I enjoyed both.
The first game on the Wii that I played seriously was, of course, Super Mario Galaxy. That was the game that got me sold on the Wii back in late '05/early '06. I still haven't gotten around to Galaxy 2, though...