When it rains it pours! I'm getting a crash course in vanagon repair!

The battery is going dead. First time it died, we put a battery charger
on it. It ran for 2 days, the third day it was dead again. I've cleaned
the battery terminals. The positive cable looks somewhat corroded. I'm
going out now to try to test the alternator. That's the only thing I can
think  of. Any suggestions? It's an 87 watercooled vanagon, manual trans.
 I'm wondering why when we try to push start it, it doesn't start?
Thanks,
Sonia J