For the Nintendo 64, it was James Bond GoldenEye, the classiest of the classy as far as first-person shooters go in my opinion. I cherish those memories of sneaking through the air-ducts and shooting guards with my silenced PP7.
For the original Xbox, the first game I played was Jet Set Radio Future, and is still one of my all-time favorite games. I used to be a huge fan of roller blades, so this mix of unique music, a great story, spectacular villains, and of course fast-paced roller blading action, made its way into being one of my favorite video games.
On the Gamecube, I played through the original Metroid Prime, which reminded me a lot of some of the Legend of Zelda games, just with a darker atmosphere and future age technology. Really great game by the way.
I also have the slightest memory of playing a dolphin game on the Dreamcast, which seemed like pretty ground-breaking technology for its time. Not too sure if I would recommend it though now.