On a Mac, the best way would be to mount the target computer as an external hard drive as mentioned, then start-up iTunes with the option key held down (let's you choose a new library to open), open the library on the target machine, then go into iTunes preferences advanced tab and change your library folder to the folder on the new machine or wherever you want the music to be stored. Then click File -> Library -> Consolidate Library (which takes all the music you have within iTunes and copies it to your iTunes music library (or whatever you selected as the storage location)... although I think the new version of iTunes has an export library function, that may be the easiest route?!
On a Mac, the best way would be to mount the target computer as an external hard drive as mentioned, then start-up iTunes with the option key held down (let's you choose a new library to open), open the library on the target machine, then go into iTunes preferences advanced tab and change your library folder to the folder on the new machine or wherever you want the music to be stored. Then click File -> Library -> Consolidate Library (which takes all the music you have within iTunes and copies it to your iTunes music library (or whatever you selected as the storage location)... although I think the new version of iTunes has an export library function, that may be the easiest route?!