For the PC, I enjoy some of the strat games like Civ IV, but they tend to be long and drawn out....hard to get enough players that have time for a full game together, and if you try to save a campaign and play later, it's even harder to get everyone together. Altho the game I played with Deed and Kuratos was really really fun - I learned a lot and managed to take over enemy cities without combat!
I like the FPS games (CS, Q, UT) but I haven't played them for years. I had Battlefield 2 and played it quite a bit, but I was playing alone and a TON of people cheated (shoot through walls, insta-snipe from 1000 yards, etc) and it really took the fun out of it. I hear that BF2142 is better about that stuff, but still has problems.
I noticed that games that require kicking systems left me at a distinct disadvantage. While I have a decent computer, it's a little old and way underpowered for all the new games that are coming out. Even on the lower graphics settings, I still lag pretty bad.
For 360? No idea. I've only ever played Halo