Is there an easy way to move my iTunes library to another computer, including all the songs I obtained from sources other than the iTunes store? I don't need to keep it on the old computer. It is my understanding that the iTunes sync feature in 9 doesn't include non-iTunes songs.
Like Christopher explained, you should just move the entire iTunes directory. Importantly though there will be an XML file inside your iTunes folder named "iTunes Library.xml". This file contains references to every item stored inside your iTunes library.
Make a copy of that XML file before messing about.
Open the XML file in a text editor you're comfortable with. Near the top of the file (was the 9th line for me). You will see a line similar to this:
You'll want to replace the values inside of with file://new/location/of/your/itunes/directory.
I'd suggest doing a find and replace on the entire file.
Open iTunes, it may take a minute but everything should work.
Locate the following folder
Copy the following folders contents to (On Mac OS X)
where the tilde "~" represents your home directory.
I'm not sure whether the library file is cross-compatible that's why in this suggestion i recommend just copying the Music folder.
Have a read over here