I am going to switch from using my Mac Mini to using my newer MacBook as my music server. I am assuming I do not have to buy a new license for my MacBook but how should I make this transition?
Correct. Your license allows you to use MC on as many Macs as you, personally, own (within reason). You can use your license on both the Mini and the MacBook.
If you want to use it on a fleet of 60 Macs for you and your entire extended family plus office mates? No, that's not cool. But for your personal machines, it is fine. If you also have some Windows machines, you'll need a Master License, though, to cover those:
http://wiki.jriver.com/index.php/Master_LicenseDo I simply download from this site the current version of MC and then set the path for the HDD that will be moved from my Mac Mini to the MacBook?
Download and install the current version. Then, restore your license onto it:
http://wiki.jriver.com/index.php/Restoring_a_LicenseThen, make a Library Backup on the Mini, and restore it on the MacBook:
http://wiki.jriver.com/index.php/Library_BackupIf the drive your media is stored on has the same volume name on both machines, then you can just plug the drive into the laptop and go.