It is more complicated than just signing in, but since you have backup discs not too complicated. iTunes tracks what music you have purchased (the only reason seems to be so that it can tell you when you have already purchased something) but you can only download it once and it doesn’t keep track of any music that you have imported from CDs etc.

What you will want to do first is go to your old computer (if you still have access to it) and deauthorize it. iTunes only lets you authorize 5 libraries to play songs that you have purchased from the iTunes store so it’s always a good idea to deauthorize computers you are no longer using. To do this, open up iTunes and under the “Store” menu click on the “Deauthorize Computer” option and enter your username and password.

Then, once you have iTunes installed on your new computer, start sticking the backup discs into the drive. iTunes should give you instructions on what you need to do to import them (I’ve done this, but it’s been a while, so I don’t remember exactly what it had me do). Continue doing this will all of the discs. Then you’ll need to authorize your new computer to play songs. This is done pretty much the same way you deauthorized your old one – go to the “Store” menu, click on the “Authorize Computer” option and then enter in your username and password.

The deauthorization/authorization only effects the songs you have purchased off of the iTunes store. Songs you have loaded from other sources i.e. off of CDs are not effected by this.