It depends. Is your old Xbox dead? If so, did you replace it with one of the new slim models? If not, just plug the old HD and the USB stick into the new console, transfer the saves to the USB stick, turn off the Xbox, plug in the new HD and transfer the saves over from the USB. It shouldn't take any longer than 5 minutes. Here's a guide
on how to transfer the files.
If your new console is one of the slim ones then it's considerably more complicated and probably involves a screwdriver, if it's even possible at all.