Technically the first game I played was Halo: Combat evolved, but it was just the first full-length game. I remember playing the Chronicles of Riddick game that was included on the DVD (might have been the Butcher's Bay one), and it really amazed me at that time. I remember hearing about ragdoll physics before getting my Xbox and I found them highly amusing.
Just like many other posters here, the first game I played on xbox 360 was Halo 3, followed by Soulcalibur 4. Both awesome games, I remember playing nonstop for hours, was going full nerd because everything was so cool.
My first game was the original Gears of War. I was at a buddy's house and he talked me into trying to use the chainsaw to kill people, and I found it so funny and over-the-top that I resolved to get myself an Xbox 360 and try it out.
I played the original Halo, but then didn't even pick up another version until 3. Three was kind of a rebirth for me. I probably hadn't been that excited for a game since I first picked up a controller.
Believe it or not but I only got to play the original xbox a couple of months ago for the first time. I was doing the gaming module of my IT course and comparing a variety of consoles was one of the set tasks. So Halo was our game of choice.
The first time in a 360 was playing on a friend's and his copy of Perfect Dark Zero. Needless to say I was pleased with the console but disappointed in the game. I picked up my own a year later after Dead Rising came out and I got an HD screen to play it on.
First ever game on an Xbox was Halo. Played it LAN with a bunch of friends the first weekend it was out. Will never forget some of the epic LAN parties from those days. We spent half of the time playing, and the other half just discussing everything that happened. Never have experienced anything like that before or since.
First game on Xbox 360 was Kameo. I really liked it. Second game on 360 was Hexic HD since it came with the console. Third game was Oblivion, still need to go back and beat Shivering Isles.
My favorite accomplishment on 360 is getting all achievements on Assassins Creed. Next favorite would be 'Mile High Club' achievement from MW2.
Most frustrating game I have played on 360, probably Lost Planet 2 because I tried playing it alone and the freaking sandworm boss fight was horribly designed for single player.