All the usual disclaimers apply . . in short, do this at your own risk, blah blah blah. Also, people have different methods of doing this job, and I have mine.
Backup lights don't work? Can't see behind you at night? If your bulbs check out okay, then the next step is to replace your backup light switch. This should take no more than a few minutes, and the switch costs under $10. This is probably the shortest and easiest DIY, but I wrote it anyway for those not so mechanically inclined 
Jack up the car to give you more room to get underneath. Photo shows the switch on the driver's side of my Getrag transmission. On ZF trannies, it will be on the passenger side towards the front.
To pull the wire harness off, feel for the metal locking clip and push in. Then pull the harness off the switch. Photo shows the harness off, with the metal clip visible (sorry for the blurry picture).
You'll need a 3/4" box end wrench to unscrew the switch (inset photo), since not much else will fit in that space. If you use a socket set, you may need a deep well socket, but it might not fit with the wall of the tranny hump so close by.
Say goodbye to the offending switch, and pop in the new one. Official torque is 15 ft-lbs, but a smidge past hand tight will do just fine. Plug the harness back in, lower the car, and you're done!