How do I transfer my IPOD music to my new MAC that now also has purchased music on it?

For iTunes purchased music

iTunes Store: Transferring purchases from your iOS device or iPod to a computer

For other music see:

Recover your iTunes library from your iPod or iOS device

Note the to sync to the iPod from the new computer you have to restore the iPod on the new computer

Use your lightning wire (charger wire) and hook it up to your Mac and iPod. It will ask you to trust the computer and then it will automatically sync to your computer if you click on the iTunes application.