92 Lumina 3.1-Battery or alternator?

I bought a 92 Lumina 3.1 about 7 months ago. I had to replace the fuel filter and pump after about a week and it ran fine until last weekend. As I shifted from reverse to drive, it stalled. I restarted it and it stalled 3 more times before I was able to get it back into a parking spot. I accidentally left the dome light on over the weekend and when I came back the battery was completely dead. I jumped it from another car and let it run for 15 mins to charge the battery. I then started driving again for about 1 minute, and the radio started making a weird noise and the speedometer was jumping erratically. I jumped it using a battery charger, and was able to get it back into a parking spot. It ran for about 1 minute before it died again completely.
Is it the battery, or alternator, or computer? I'm new to Chicago and don't know any reputable mechanics, so I'd like to have some idea before I take it in. Any help is appreciated.

Yea That Sound Like Its The Alternator.
Because If You Have To Keep Charging Your Battery or Getting It Just stared When That Happen Something is killin your Battery And Your Alternator is not charging it so try to go with a new or used Alternator

look at left fuse box under hood you will see post on it make sure it is clean and tight

take the battery out and charge it till it has a full charge.
if it starts and runs for a while. and I mean longer than moving it to a parking spot. you should be able to go at least 5 miles. than the battery should be ok.
once you get it started use a volt meter. hook it up to the battery it should read about 12 volts. reeve the engine with the volt meter still attached it should go up to 13 or 14 volts.
if it does not then the alternator is bad.
if both of them check out ok get a codes reader and check the computer.
if the car runs once you charge the battery drive it to the nearest auto zone or advanced auto and they can check all that for you for free. and save you the trouble