If I bought the game a while ago and get a different computer, will I have to buy it again?
I think the D3 license is attached to your Battle Net Account, not to a computer. As long as you remember your Account username and password you can log in from a different/any computer to install and play the game. You can keep using the same D3 copy you bought years ago, dont need to buy a new one to play on a different computer.
Nope.
Download and install the Battlenet client onto your new PC.
Use the Battlenet client install Diablo III.
Log into it and play it.
all your acct info is stored on Blizzards end its the same as buying a digital movie from Amazon you can simply route it to your new devices etc