Unfortunately, you won't be playing just yet. You'll have to download the latest patches from the Blizzard downloader (a Bit-torrent sort of thing) which isn't all that fast, so I'd recommend finding an FTP or HTTP mirror of it.
Once that's done, the fun can start
Choose either PvE or PvP server - PvP means whilst you're out running around at level 20+, you can be attacked and killed by players from the other faction. However you can do the same to them should you so wish to. On a PvE server, you can't attack the other faction unless you both consent to it by typing /pvp. Both types of server do have Battle Grounds which you can enter and fight in mini games of Capture the flag, or hold the bases.
There's also RP or RPPVP server types - these are for the hardcore Warcraft bod's
Then choose a faction, a race, and a class. This is entirely up to you, but easy classes to level up are hunters and rogues.
After that, you're ready to go, find yourself a good guild with some new players in and you'll learn the ropes fairly quickly.
Bare in mind though, that nearly every server has more Alliance players than Horde. This means more players to quest with, but Battle Ground queues will be longer.
Edit: Also, it'll probably help you a bit if you download some add-ons to improve the standard User Interface. I'd recommend downloading 'Titan Panel', 'CTmod', and 'Scrolling Combat Text' to begin with. You can grab them from
www.curse-gaming.com. Unzip the files to X:\WorldofWarcraft\Interface\Addons and you're ready to rock
