i did something similar not that long ago....
trying burning the music onto a dvd-r/dvd-rw, you'll get about 4gig on 1 dvd, which is a fair few songs. If not split them up and put them several dvds.
=OR=
As previously mentioned put them on an external hard drive / memory stick. If you cant fit all your songs on a memory stick, transfer what you can on to it. Put the music files onto the other computer, deleted the files on the memory stick then add some more music from the original computer.
hope this makes sense, had a little too much to drink and cant sleep!