Xbox Live is still up and running but I'm not sure how many players you might be able to find. I know I'm #1 on the weekly leaderboard for XIIZeal and DeltaZeal... I think I might be the only player worldwide to play that game in 2017.
My biggest recommendation for the 360 are shmups. Deathsmiles is one of the best and you can find it very easily for cheap. Akai Katana's another great one that's also easy to find. I also love Eschatos, but that one's a Japanese import (but is region free).
One bit of advice: if you can at all find a new, official wired controller, snap that up! It looks like Microsoft has discontinued them. I thankfully was worried about that happening and got one a while back just to have as a backup. It's the most long-term reliable controller.
The console came with a wireless "play & charge" controller, but the battery seems to be 100% dead (there's also an adapter to use AAA batteries, but I don't want to be burning through those) so I just leave it plugged in all the time.
