Did the AAP person charge the battery and then perform a load test? Do you have a battery charger? Sometimes if you charge a battery and it does not accept the charge, it could have one or more dead cells, which means the battery needs to be replaced.
However, if the battery can be charged to beyond 12 volts, and you can successfully start the car, then you should measure voltage while the engine is running. If the alternator is good, you should see over 13 volts. My car alternator outputs about 13.5 volts.
Make sure that the battery cable clamps are pushed down all the way before they are tightened. The connector's inside diameter and post should be cleaned with a wire brush tool that is made for this purpose. Once each clamp is tightened, you should not be able to rotate it.