Make sure that you get a USB or FireWire hard drive that is formatted to work with Mac. The format to specify is Mac OS Extended Journaled.
Connect the hard drive to the MacAir using the USB or FireWire cable and drag the files that you want to keep on the new hard drive over from the Mac's hard drive.
Click the Time Machine "clock" icon at the top of the Mac's screen
Click Time Machine Preferences
Click Options
Here, you will likely see that the hard drive is on the lost of "excluded" items. You do not want the hard drive to be "excluded", so click on the name of the drive to highlight it, then click the - (minus) button at the bottom of the list to remove the drive from being excluded.
Time Machine will now backup your Mac and the hard drive going forward.
Once you have verified that the files are on the hard drive and that they are being backed up, you can delete the original copies of the files on the MacAir to free up space.