You could do this a couple of ways depending on your computer expertise and how much you want to spend.
1) Buy a USB external CD or DVD burner, hook it up to your old machine and burn all the files to disk for transfer (And then you also now have a copy of them if the new system ever goes down). You could also buy an external USB hard drive and transfer all the files to it. These are the most expensive methods as you would have the cost of the external CD drive (+cd's) or hard drives, but you would then have back ups of your data, which is never a bad thing! You can also use either of these devices on your new computer. Should all be "Plug N play" (Unless you are running win98, then you will need a device driver).
2) Buy a USB thumb drive (at least 512 mb) and transfer the files that way. Kind of a pain in the ass, but it would work. These are running about $70 to $100 right now. They work pretty good and you can also use them for future file transfers. Also should be "Plug N play (see above re: win98).
3) If you are not afraid to crack open your new box, take the hard drive out of the old machine, configure it as either a slave or cable select (jumper pin settings on the back of the drive) and install it in your new machine. The new machine should "Auto detect" the drive from you old machine and install it automatically. Then you can transfer the files from your old drive to your new one. This one won't cost you anything, but you need to make sure to configure it correctly or you could lock up the computer. One thing about this method is that breaking the seal on the new case (if there is one) may void the warranty, depending on where you bought the new system from.
IM sent.