You run MA on the new Mac.
If you really want to just move iTunes:
iTunes: How to move [or copy] your music to a new computer [or another drive] - http://support.apple.com/kb/HT4527
Quick answer if you use iTunes' default preferences settings: Copy the entire iTunes folder (and in doing so all its subfolders and files) intact to the other drive. Open iTunes and immediately hold down the Option (alt) key (shift on Windows), then guide it to the new location of the library.
You run Migration Assistant on the new computer. The old computer does not need migration Assistant.
Use the Migrating using FireWire or ThunderBolt link on that page.
& the new one is running Lion or Mt. Lion.
If it's new, it's (should be) running Mountain Lion (10.8). Go to Apple menu > About This Mac.
if after transferring music from one computer to another or after re-downloading purchases to a new computer playlists may not be saved. there is a simple way to re import playlists to the new computer or the computer with out playlists (Note: this does not import songs only the organization of the songs in the playlist)
- plug the device that has the playlists into the computer
- on the side bar click the arrow to the left of the device
- highlight the playlist you want to export to the computer and right click if on windows control+click if on a mac select export playlist
- save as (playlist name).xml save to easy to access location such as the desktop or documents folder
- in itunes select library from the menu bar and import playlist
- import playlist file that was just exported to the computer
- after importing your playlists will be as they were on the device