My Celebrity 88 won't start :(

Hi.
My car is: Chevrolet Celebrity 1988 V6 2.8L

Couple of months ago, my friends tried to install a new radio with CD player into my car. The new radio worked fine, but the rear lights weren’t working and the left turn signal did not dim, but was constant. Looks like they tampered with the wrong wires.
When winter began, my car wouldn’t start anymore. I removed the radio, jump started the car. Now, all lights were working fine, but as I removed the booster cables from my car’s battery, the car silenced and engine stopped working. One of my friends who is a mechanic looked at the car and said that I either need to purchase new:
- Spark plugs
- Battery
- Alternator

He suggested I first purchase a new battery.

I was waiting for winter to end and for the temperatures to become warm to start working on my car. So yesterday, me and my buddies jump started my car again. The ignition did not work right away, but we were able to ignite the engine. We left the enginesof the two cars running for about 7 mins. I disconnected the cables from my car’s battery and the car was still running. I was all happy. But, when I pressed on the gas pedal, the car died again. We tried a 2nd time with the booster cables, the engine was running by itself again, but when I opened the door of my car and the light inside the car ignited, the car died again.
I know that the engine is in good shape though. So my guess would be that the battery is dead and probably got screwed when the radio was installed badly. But the car was still driving (without the back lights though) with the new radio.

I was wondering if u guys know what the problem might be?

Thanks a lot!