The two most common breakdowns for garage door openers are:
1. Broken door lifting coil springs
2. Worn motor drive belt
The springs are simple but a little tricky to replace. You'd need two 1/2" steel rods, about eighteen inches long each. They are used to tension the springs - about seven turns each. Exreme care required here, as the energy in the loaded springs is very dangerous if they let go accidentally. It takes a little bit of elbow grease, but if you don't feel up to the job, then you should hire someone. The springs themselves cost about $40 each.
If your motor still operates, but the door doesn't engage, then the belt may be worn or broken; a simple replacement. A new belt will set you back about $15.
Most other problems involve replacement of the motor or the whole system, which can be had for about $250 at Home Depot or Sears.