I know you can transfer licenses in Office 2013, but I'm not 100% sure how to do this. My main question, though, is how would I get office 2013, or authorize it on the new computer, since I got this version through the upgrade program after buying Office 2010 at the end of last year?

Simple:
Reinstall Office 2013 on the new PC and, when prompted for the qualifying upgrade product, insert your Office 2010 disc into the drive.

__________________
Cheers,
Paul Edstein
[Fmr MS MVP - Word]

So, I just re-download Office 2013 (or, if it already comes on the computer, open that up), then put in the Office 2010 disc?

Having not installed Office 2013 (too buggy for my liking - I'm staying well away from it, as are various other MVPs), I can't be 100% certain about the process.

What I do know is that the free upgrade, which is what I assume you're referring to, was only for Office 2010 users who bought & installed their software during a certain period. You may need to reinstall Office 2010 before installing Office 2013 as I can't see any reference to the kind of upgrade version MS used to produce. Even then, I can't be sure it'll work as MS hasn't published anything on this aspect. Conversely, you may only need to supply the Office 2010 product key when installing Office 2013. Alternatively, you might use a product like The Magical Jelly Bean Keyfinder (http://www.magicaljellybean.com/keyfinder/) to retrieve the Office 2013 product key and, if it differs from the Office 2010, use that for the new installation.

__________________
Cheers,
Paul Edstein
[Fmr MS MVP - Word]