This list is probably not in actual chronological order, I have a fuzzy memory. :p
The first game I played online was probably the Ragnarök beta with my brother. But when he quit, I later quit too; it was no fun anymore. Just grinding and talking to the occasional cool person among a sea of immature morons and racists. One of my characters were female. I'll never repeat that mistake.
We played a lot of Counterstrike with friends, but not that much on public servers. I remember getting killed by campers of the wrong team (e.g. Terrorists are not supposed to camp their own starting point in bomb levels, and CT not in hostage levels) and having to wait a few minutes to get back into the game. I remember one guy who was invisible; his argument was that it was his birthday, so he deserved a day of fun ruining other people's games. Was fun looking at some pro-level matches though. I remember one where a member of NIP just barely won the match by throwing a flash around a corner, then jumping in and shooting the guy, who was now turning his back, before the flash went off. Or mayeb it was a smoke that he knew the other guy would think was a flash... nice move nevertheless. :p
Me and my brother later played some Rocket Arena in Quake 3, the most balanced and fair FPS mod I know. Didn't last too long though.
I played some Deathmatch Classic (Quake 1 in the Half-Life engine), but that consisted half of getting kicked for owning people who still play with the arrow keys, and half getting owned by real old-school Quakers. No fun.
I started playing Tibia with a friend on and off for a while. This was a few years ago. Tibia must be the worst MMORPG ever invented, unless you're a sadistic powerleveler. Grind, get lag-kicked which kills you and drops you a level, grind. Snore.
Tried some Maple Story on my own. Grind, no fun...
Most recently played game was Ragnarök on a private server, 3x the experience and skill. People were cool there, and you didn't have to slave for days to get cool abilities and fight new monsters. But in the end, it's just grinding too. At least it's not Tibia.
Never played a strategy game online, I suck at them. My friend used to insist on playing Total Annihilation, and he won despite whatever handicaps he gave me. I think I only won against my brother in Warcraft 2 once, when he forgot to upgrade the armor and attack of his troops. As for Starcraft, everyone has a personal army when I've just started setting up a few defense towers in my first - and only - little base. :p