Grab the full client from the Download section of the menu. It already includes the HD graphics pack and every update, so there's nothing else to install.

Download the client

Extract the file into the folder where you want to keep the game. There's no installer: once it finishes unzipping, the game is installed.

Avoid Program Files and the Desktop. Something simple like C:\Games\WoW works best and saves you permission headaches later.

Open the folder where you unzipped the game and go into:

Data\enUS

Open the file realmlist.wtf with Notepad, delete whatever is inside, and leave only this line:

Save the file and close Notepad. This is the step everyone forgets, and without it the game has no idea where to connect.

Run Wow.exe from the game folder and log in with the account you created on this site.

Use the same account name and password you signed up with — not your email address.