How do I move GarageBand to a new Mac?
If iLife came pre-installed on your computer back in 2008, your computer came with one or two install discs. If there were two, iLife installers exist on the second disc.

You didn't specify what kind of computer your old or new computers are, but if they both have FireWire ports, you could use Target Disk Mode and a FireWire cable to transfer files faster than via your slow network.

A third method to transfer GarageBand (and other applications) from your old computer to your new would be to use the Migration Assistant application. You can't select individual applications to copy, but you can fairly seamlessly transfer applications from one computer to another, then delete what you don't want.

Without knowing any specifics about hardware or OS version, it's difficult to give you better advice, but your three best bets to more seamlessly copy the application are 1) the install disc, 2) copy the files using Target Disk Mode, or 3) use Migration Assistant.