Yes, that's correct. It's relatively easy to move your itunes music from one computer to another, and I'm pretty sure that was one of the OPs questions.

sdsvtdriver is also correct in a sense. If you have two computers with itunes on it, and haven't copied the itunes music library over from one to another, itunes on the new computer will recognize that and present a pop-up warning you that if you continue, your music on the ipod will be erased. Obviously you hit cancel at that point. itunes will only copy itunes purchases from the ipod into itunes if the new computer is "authorized". He said "itunes won't do it", and that's correct, but slightly misleading because you don't do the copy using itunes itself.

Clearly you want to manually copy over your itunes library and music from the old computer to the new computer before you start using itunes on the new computer.

There's also a good article on ilounge in the ipod 101/201 series of articles that describes other ways of moving the music to the new computer, including getting the music from the ipod's internal directory structure itself.

-David