Many people my age remember playing games on xbox 360, but my memories are different and mostly even older games.
I first remember playing games on my brothers PC when he first got it around 2009 or so. This had a week Celeron and my parents didn't want us on the internet or something, so we ended up running games from early 2000s on CD.
We had a 1000 games one CDs and other cds, so its not really a single first game.
One was called wild wheels. You drove around rc cars to complete races and missions. I distinctly remember a level where you had to deliver bombs to defusal places in time. I think i remember this since it was so hard.
I also remember a game called speedy eggsbert. It was a puzzle game where you could complete obstacle curses or even design your own. I remember it crashed a lot too, probably a windows vista thing.
I picked up a copy of NFS 2 SE at a thrift store and the fun times i went through to get it working on vista. Not fun. This game made me love the McLaren F1.
I also remember a game called petz. This was on like a wendys kids meal cd. You got to have dogs, play with them, train them, and take care of them. Your whole objective was to win talant shoes.
I remember a game called funkez. You had a city of sorts and you could do many things, but i cant remember any specifics other than monjong tiles. You could customize your little creature and room too. The game had little plastic characters with codes.