The migration process can take quite a long time because it gathers up thousands of files. If you want to transfer just selected files, you might want to consider these other options.
You can use a Zip drive with a USB connection, a portable hard disk with a USB or FireWire connection, or a floppy disk drive that connects to your new Mac. Just copy the files onto the disks as you usually would from your old machine (drag them from the hard disk of your old Mac onto the disk icon). Then connect the drive to the new machine and copy them onto your hard disk. If you have a very old Mac, the trouble, of course, is finding drives that work on both an old Mac and the new one. If you have a drive for your old machine and a different drive for your new one, the actual disks should go between the Macs just fine.
If your old machine can burn a CD (or a DVD) you can transfer quite a few files that way.
If you don't have huge files, send them as attachments through email. It's kind of funny to send your files around the world to get to the machine sitting two feet away.
If you have a .Mac account, put the files from your old Mac onto your iDisk (see Lesson 11). On your new Mac, download them from your iDisk.
If both machines are on a network, connect the two computers and copy files directly from one to the other. See Lesson 10 for some minimal details.