I never played World of Warcraft before. So I thought I will give it a shot and tried to download the 'Starter Edition' which lets you play for free up to level 20 from my Battle.net account. But after clicking the 'Game Installer', it only downloads the client. It doesn't even start the launcher. So I saw an Youtube video on how to download the Starter Edition. It shows that the game launcher automatically starts after downloading the client. But in my case, it doesn't happen. The Battle.net account says-"Oops, sorry. We couldn't find what we wanted to show you." So what to do? I live in India. Is the game playable at all from India?
Assuming I understood your question correctly, you got the first steps right; Log into your account in their website and search for the menu to download game clients. There you will find the WoW installer (which will actually install the battle.net launcher) and then you simply install it.
After installing it, you will now be able to open the battle.net launcher (which requires you to log in with your e-mail and password) that enable you to actually download and install any current Blizzard games, including WoW.
To my knowledge there is no restriction on Blizzard's end in any of these steps to someone playing from an India IP, though local legislation may still apply.