Looks like your computer might be a bit old to support USB (and USB flash drives)... so the best you can do is follow William's instructions, however, William is a Pro (i.e. he knows what he is doing and he is supposed to do) so you might feel a little uncomfortable in taking the hard drive and putting it in your new computer.
So here's my solution (a little hard, but it should work):
Take the new computers CD burner (if you have one in the new computer, if not I suggest you might want to get one...) and put it in the old computer, then burn all your files. And then put it into your new computer... changing the CD burner is fairly easy and take a look at this thread
if you aren't sure how to do it...
Hope this helps