How can I tell if it's the battery or something else?

I would say you have a bad battery, if the alternator was not charging the car would probably not idle for a half hour and the "battery" light would be on. There is the chance that the alternator is over charging and ruined your battery. Take the car to a parts store or shop and have them test the battery AND the Alternator. You will probably need a battery either way but you dont want to ruin your new battery on the SLIM chance your alternator is over charging.