Assuming you have a 100mb network - I agree that transferring data through the network would probably be the easiest. This route, you don't have to touch 1 hard drive.
If you didn't have a 100mb network and your computer has USB2 and you can do your enclosure (or usb to IDE), that would also work in decent time.
If you didn't have either a 100mb network or USB2, than moving 1 hard drive from one computer to another would be ideal. 1 thing about jumpers though. I don't know why everyone always installs the second temporary hard drive on the master channel as a slave. Simply unplug your optical drive(s) and use an IDE plug off the secondary channel. No jumpers need to be configured and no risk of things going bad with your system drive.
I would expect transfer times for 20gb to be(assuming 7200RPM drives):
100mb network - 20 minutes
USB2 - 20-30 minutes
Slaved Hard Drive - 20 minutes
10mb network - 10 hours
usb1 - 10-15 hours
Good luck. Your best bet is to pick a method based on time and than learn how to do it.