Granted that people are naming non-Mario games for non-Nintendo systems, I shall join too.
NES- Named in my previous post
SNES- Gradius III, the Gradius franchise is full of top notch shumps, and III is probably the best in series. I have lots of history with that game. I never managed to beat it, but hey, it is a fairly challenging game. To this date it's one of my favorite SNES games. I played it before I even played any Mario SNES games. I also had Turn & Burn: No Fly Zone in that time, but that game was horrible, so I didn't play it more than a few times. Not long after, I got Yoshi's Island, which consumed tons of my time.
N64- Diddy Kong Racing. Of course, my first games played were Super Mario 64, and Super Mario Kart; as they struck my interests more, I played DKR not long after. I loved that game so much. To be honest, some of the elements in it were better than MK64. Better battle tracks, bosses, definitely better story line. Of course, MK64 was easily better and had consumed more of my time. But DKR was definitely played a lot.
GameCube- Metroid Prime. My first games obtained were Super Mario Sunshine, Super Smash Bros. Melee, Godzilla: Destroy All Monsters Melee, and some crappy Hotwheels game that I ended up selling. In addition to Metroid Prime. I didn't get SSBM, Godzilla, or Hotwheels until later that day. So my first two games played were Super Mario Sunshine (which, may I add, is under-rated), followed by Metroid Prime. I really want a Wii so I can get Metroid Prime 3.
Wii- I am sorry to inform people, I do not yet have a Wii, and the only game I played on it was a demo of Super Mario Galaxy. So I'll get back to you with the first non-Mario game on the Wii. Probably Brawl, since I plan on getting Galaxy and Brawl as my first games.
Game Boy- Generally the obvious answer; Tetris. I only had Super Mario Land and Tetris at first, so naturally I had played Tetris as my first non-Mario game.
Game Boy Color- Pokémon Crystal. My first game was, I believe, Wario Land 3. I didn't really buy many GBC games, since I never really got the opportunity to purchase any that struck my interests. So Pokémon Crystal, which I got years after my GBC, was my first.
Game Boy Advance- Another sluggish one. I had Super Mario Advance (SMB2US) for a while, then I lost it. About three years later, I bought Driver 2 Advance (God awful). Then my GBA library began to expand.
Nintendo DS- Metroid Prime Hunters: First Hunt Demo :P