1998 Chevy Monte Carlo alternator?

I have a 98 Chevy Monte Carlo which has about 120,000 miles on it. It has been running very well till a couple of weeks back.
Since I have a newer honda civic, this car has been idle for a couple of weeks. One day, when I took it to the nearby gorcery store, the ANTI-LOCK indicator came on. The brakes worked and it ran without any problem. The next day I noticed that the pickup(acceleration) wasn't very good (actually very bad) and the VOLT indicator also came on. The next day the car wouldn't start. I used my honda to jump start the car and it started. I went for test round inside the parking lot and the car went dead mid way while it was running. I somehow steared it to the side and again jump started it and got the car parked.

A couple of weeks later (ie. One week before), I bought a new battery and replaced the old one. The car started and ran without any problems. But it was short lived. Within a couple of days the VOLT indicator came back again.

1) What could be wrong with the car? This looks like a an electrical issue to me since it worked well after changing the battery. What I would like from the experts is the following:

2) Could this be a problem with the alternator? I have read a couple of similar stories in the internet and the problem didn't go away by changing battery/alternator. What could be the chance for that?
3) What parts are involved in the issue? Why was there a difference in the pickup (acceleration) after changing the battery? I was under the impression that batteries and associated parts are not involved that much once the car starts moving.

Thanks.