2002 Cavalier idles fine, stalls when driven

I’m a rank amateur trying to diagnose the problem with my 2002 Chevy Cavalier (2.2L, automatic, 101k miles, no-frills aftermarket CD player). It drove fine until one day it just wouldn’t start. I gave it a jump, and got about half a block before it died again. Jumped it again, got another short distance before it completely died – not even enough power to run the hazard lights or power steering. I called a tow truck (since my jump-starter was drained) and he was able to start it and left it idling while he towed it back home (low battery plays havoc with the anti-theft). An hour after I got it home, I was able to start it normally. The next day, the battery was completely drained – not even the dome light would come on.

I can still jump start it, and it hasn’t died in the parking lot, but I haven’t tried driving it again (for obvious reasons). I got a cheap multimeter, and the voltage fluctuates between 14.83 and 14.89 while idling (I’m new at this, so I don’t know if that’s normal, or if it’s just the multimeter, or if it’s a bad thing).

The battery is a 2-year-old ACDelco (still under warranty). I believe the alternator is the original.

My plan is to take both the battery and the alternator to be tested. I want to check for parasitic drain, but I need a decent battery charge to do that. What else should I be doing? What else should I look for? Any ideas as to what the problem might be?