Just to help a bit on the saix side:
games.saix.net lists the games for which they have servers running. So you buy a game, ensure that it's patched to the latest patch level, and then use the above saix site to get the IP address and port number of the server you are going to be connecting to. Then you enter the game, go to the multiplayer/online settings, and either look for the server or enter it in manually (eg: favorites). When you are connected, you'll enter into the game, with other around the country playing as well.
That's from the saix side, but there are other locally hosted games servers. Some games only have international servers and no local ones, which means that you'll generally be playing with a higher ping (which means a slower response). This isn't so bad for some games, but others like FPS's you'll want as low a ping as possible.
Best to decide what type of games you want to play, then find out what everyone is currently playing in that genre, and on what server's they are playing. Then go out and BUY the game if you don't have it - more than likely you won't be able to play online with a cracked game, especially if punkbuster is required to be running on your machine as well(in some cases it is possible, but then you are most likely playing on a cracked server). Either way, you'll then get the most enjoyment out of the online gaming, because then there will actually be people online and playing with you...