Age old question...Bad Battery or Bad Alternator???

What car do you have that has a voltmeter and what type?
It could take more than a 10 mi drive to recharge a battery. Put it on a charger until full. Autozone and others won't test a battery unless it is fully charged.
You should see >13 V at the cigarette lighter with the engine running, especially on the highway.
I have a Black & Decker 10A smart charger with built-in "alternator test" function. I understand it simply looks for a sinusoidal fluctuation, indicating a live alternator. Got it cheap at Big Lots. For a better measurement, I use a clamp-on DC ammeter probe (F.W. Bell, $15 used on ebay).