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.
I used to help my mom on dungion siege, but that technically wasn't mine. I would have to say looney tunes back in action. It was an extremely fun game at the time and it still would be, if I hadn't of scratched in on accident while putting it in the PS2.
The original Super Mario Bros. on NES. The one that came with Duckhunt on the same cartridge. Yeah, I'm an OG.
I am just going to figure that the intent of this thread was to say what the first game you owned was. Otherwise, I'd had to look deep into my memory to recall some of those old arcade games. XD