You must be logged into the account from which you made the purchases. Hopefully you have not been using multiple AppleIDs in the past or this will take a long time (90 days wait between re-downloading from different accounts).
Downloading (using iOS or computer) past purchases from the App Store, iBookstore, and iTunes Store - http://support.apple.com/kb/ht2519
You would be much better off transferring your library from your old computer.
Moving a library: Quick answer if you use iTunes' default preferences settings: Copy the entire iTunes folder (and in doing so all its sub-folders and files) intact to the other drive. Hold down the option (alt) key (shift on Windows) and open iTunes. At the prompt to create or choose a library choose the copied iTunes folder.
When this is to a new computer put the copied iTunes folder in the default location of Macintosh HD > Users > *User Name* > Music then you don't even need to start with the option key held down, iTunes will automatically look for it there. (Make sure there isn't anything already in the iTunes folder there that you want to keep since you will be replacing it with the one you are moving.)
