How do I safely replace a worn out electrical outlet?
If you plan on doing much electrical work around the house, get a multi-meter. It's just a little box with black and red probes on the end. Safe enough to stick in an outlet to see if it's live or not.

Outlets are mostly idiot proof. Turn off the breaker, pull it out of the box, see where the old wires are attached and move them one by one to the new outlet. Probably the old outlet already has a ground wire, in which case you're done. If it doesn't, it's probably time to call a pro.

Once you have it hooked up and the power is back on, you can use the multimeter to test. There is a hot side and a neutral side to the outlet. Ground behaves the same as neutral as far as the multimeter is concerned. You should be able to stick one prob in the hot side and the other into ground or neutral and get around 110 on the multimeter (+/- 2-3 volts).