It could also be an alternator issue, and would be the easiest to diagnose.
Take a voltmeter, take voltage mesurement, start the car, check the voltage again.
The interresting bit is mainly the second readout.
The first would help to have an idea of the battery state of charge. Far from being precise, but sometime can show a bad thing.
When the engine is running, do you get atleast 13V? 13V is low, but would indicate that the alternator is atleast partially working (a full battery would be max 12.7V, due to the small loads here and there would show up at about 12.5V)
Ideally you would see 14.4V, but 13.5 and up would indicate that it charge.