First of all, the concern about that alternator 'test' is because it might produce voltage spikes that might damage electronics.
Your electrical system should have been checked before the battery was changed. The tests are simple...... use a digital volt meter and attach the leads to the battery. A good system will measure approx 12 volts when the engine is not running. Next, start the car. The voltage should increase, and read between 13.7 to 14.5 volts. Any less means the battery is not being charged properly.
Your dashboard volt meter gives you this information, too. Unfortunately, these gauges are pretty inaccurate so when your dashboard gives you unusual readings (as yours is now) you need to follow up with a more accurate volt meter.
A battery can be charged and load-tested quite easily, to see if its good. If your new battery is bad (unlikely) it can be changed under warranty.
Poor electrical connections at the alternator and/or battery can cause these problems.
A severe electrical drain as the above poster noted, can also be tested for, quite easily.
Did the Autozone people tell you to change the battery in the first place? Don't go back to that particular location as they are giving you bad advice. Find a good independent mechanic (NOT the dealer - they are too expensive) and go to him. Try asking your friends/relatives who they use, if you do not know an honest one.
Finally, in the old days, a bad V-type belt could cause a no-charge and no-start situation because it slips and does not turn the alternator enough to produce a charge.
But then newer serpentine generally never slip under normal use, even if they are about to break.