My first real online game was Star Wars Galaxies, which I played from not long after it came out and until the CU hit, which killed the game, at least for me. Pre-CU though, it was awesome, being a true sandbox MMO, unlike the theme park variety that a lot of MMOs are today. Changing your profession "on-the-fly" and not being bogged down, once you made your choice was fantastic, even if I stuck with being a Tera Kasi Bounty Hunter for most of my playing time.
My first non-MMO online game was Battlefield 1942, which was not that long after I had quit SWG. My brother and me bought the game, began playing online and was completely hooked, both of us being in clans and doing clan wars and all that, until BF2 came and a lot of players migrated to that game instead. It was amazing while it lasted though, it still being some of the most fun I've ever had in an online FPS.