1998 Ford Escort car will not start

For about a month when I started my car, a dummy battery light came on and the car had minnimal power to headlights, heater blower. Car ran for a few minutes then everything would go full power. Recently the car died over night. It's been cold outside. I can jump the car and it seems to run fine but no matter how long it runs the next day it is dead. It fires right up after it has run awhile but not over night. I had battery charged by a local tire dealer and they said it took a charge but next morning same thing. No start. I plan to test alternator next.
battery, alternator, what else could be involved?