If you can get it to an Autozone they can test the Alternator very easily, usually without removing anything.
It sounds to me like your battery is not charging, or not holding a charge. Where your car sometimes starts, but then dies, it sounds like the alternator isn't producing enough electricity to operate the car.
I'd start there, at least. Alternators range from $70 - $200 at a parts place, in my experience.