My thermostat works, furnace fan works, but the AC unit outside only buzzes with no movement from the fan.

Sounds like you need a new capacitor.

http://img.diytrade.com/cdimg/377701..._Capacitor.jpg

Located in the outdoor unit. Turn the power off to the unit and inspect this for a burnt wire. Or if the top is swollen or bubbled out in any way.

The fan motor not working

Air conditioner fan outside won't work, motor hot