I looked around a little and found this link to a You Tube video on how to solder:
A couple of pointers: The wire size in the video can be soldered with the Radio Shack $10.00 iron. Use heat shrink and you can use the side of the iron to shrink the tubing since you probably don’t have a heat shrink gun. If you have several wires in the harness, please stagger the joints so you don’t have a giant wad of wires on each end of the repair. If you are doing this outside in the wind you need a much larger iron and I would recommend a propane torch with soldering tip, just be very careful and practice with the heat settings. You can get one from the home store plumbing section.
Eric has a soldering video (sort of) that is worth viewing also for background info.