Apple's iTunes actually makes it quit easy to move your entire library from one computer to another. Why would you want or need to do this? I find that sometimes clients want to begin syncing their iPod or iPhone a different, newer computer. Perhaps they are moving from Windows to a new Mac.
The secret lies in the iTunes folder. Everything that uses iTunes is normally held within that folder. This includes music, apps, books, the iTunes Library database and more. Copy your iTunes folder to an external hard drive, or directly from the old machine to the new using your network connection and file sharing.
Place the iTunes folder in the expected location for all iTunes files. Where is that? Here is a trick to find out. Download and install iTunes on the new computer and then run it for the first time. This creates a new, empty iTunes (or iTunes Music) folder, depending on the version of iTunes. Quit iTunes. Find that folder, which gives you the proper location, and then replace it with the folder from your older computer. Run iTunes again and you should see your entire iTunes Library exactly as it is on the older computer. You should also be able to sync your i-devices without any odd messages about "this iPod is synced with a different iTunes Library" nonsense. The iPod should see the new computer (actually the iTunes Library) as the same one it was already using. As long as that matches, it doesn't care if that library is on an entirely different computer.