How do I move a music collection
This is pretty straighforward, the steps are essentially:


Open iTunes and consolidate your existing library (File > Library > Organise Library)
Quit iTunes
Copy your ~/Music/iTunes folder to the external drive
Open iTunes while holding down the Alt key, which will prompt you to select the new library
When you're sure it's working, delete your original library.


For more details on each step, see this iMore article. And as always, make sure you have a working backup before you start.