If both computers have a network card, you can use a "crossover cable". They sell them at places like Office Depot. They look like a regular network patch cable with male RJ14's on each end, but they are wired differently so you don't need a hub.
If both computers are setup with file sharing enabled it's a snap. If not, it can get a little more technical.
Every time I upgrade computers the new one has so much larger a hard drive than the last one, my favorite trick is to just create a directory on the new one called /OLD and drag and drop the entire C-Drive over into that directory on the new computer. Then I just move stuff over as needed. After a month or two, I kill off the /OLD directory figuring that anything I haven't dragged over must not be needed.
Another thing is to just keep the hard drive from the old computer on a shelf for a while in case something critical comes up missing.