Hey Raven! If you haven't done so already, I'd recommend downloading and installing the Battle.net Desktop App as it's now recommended to play WoW. Let me know if you have any questions about that!
Edit: Launching the game without the Desktop App will prevent ongoing updates until the game has been launched through the App.
Hello, welcome to PC!
Steam, while certainly the biggest platform for games on PC, is not the only one. Some companies are big enough to sell their games outside of Steam and have their own software to buy, download and launch their games. Blizzard is one of those companies. Blizzard's games you can get on http://battle.net . Blizzard also has their own software for their games, and it is called Battle.net Desktop App. You Can download it on http://battle.net/en/app/ . Blizzard sells their games online exclusively on battle.net.
If you want to play campaigns, you want to buy all expansions, and start with Wings of Liberty campaign. If you want to play multiplayer only, then you want to buy Legacy of the Void only.
Adding to this, here's more information about the Battle.net Desktop App, through which you'll be able to download and update your various Blizzard games. The way WoW works now... the game and (most) of the expansion data is packaged as one big bundle, and players have access to that content which has been unlocked via registration codes. Cataclysm, along with all expansion (with exception to our newest, Warlords of Draenor) are included and unlocked so long as the WoW license has been upgraded past a trial.
Let me know if you have any difficulties getting the account issue worked out! :)
Great info :) To supplement this - the installer mentioned in (3) will install the Battle.net Desktop App by default. If the game is launched without the App, it won't check for regular updates.