Probably the most straightforward way to get iTunes onto another computer (and presuming the other computer is also a Mac) is to network the two over ethernet and copy your iTunes folder across. This works fine if you have been using iTunes to completely manage your music and all your songs are in iTunes Music not in different folders. If the songs in your library are not all in the iTunes Music folder you can consolidate them. When you consolidate, any songs in the iTunes Library that are not in the iTunes Music Folder are copied to the Music Folder and the Library is updated to point to those new files. Files already in the iTunes Music Folder have no changes made to them:
Copying all your iTunes files to one location - Mac
OS X 10.4 - Connecting two Mac computers using Ethernet
OS X 10.5 - Connecting two Mac computers using Ethernet
Another way is to put your original computer into target disk mode and copy the data over Firewire:
How to use FireWire target disk mode
Copying the whole folder brings with it the iTunes Music folder and also the iTunes Library (iTunes Library.itl in Windows) database file which holds all the information about your songs (Playlists, Ratings, Play Counts, Last Played, Date Added, etc). You'll get more information including the Folder/File structure in this article:
What are the iTunes library files?
When you copy the iTunes folder make sure you put it into the same location on the new computer. This has to be the exact same place, on a Mac this would be in Username/Music/. Everything has to be exactly the same on the new computer including the Mac user account name. When you open iTunes on the new computer it will be the same as on the original one Songs, Playlists, Play Counts, Ratings etc. You should also use the same version of iTunes on both computers.
If you have and iPod, some iPods can be used as an external drive and you can use them to move your iTunes (this option isn't available with the iPod Touch):
How to use your iPod to move your music to a new computer
If you have any iTunes purchases in your library don't forget that you'll need to authorise your new computer to play them:
About iTunes Music Store Authorisation and Deauthorisation
