Starts but shuts off quickly

I was told by owner it runs but has a short or something that is draining the battery. I bought new battery and started car and it runs great. I drove it about 1 block and put in reverse to turn around and it quits running. I continued trying to start the car and it turned over just fine but then I would hear a click and in one turn the key off and continue trying to start. No luck so I parked the car and next day The car starter right up and I drove it 1 block and parked it where I bought it. This time it didn't dive I backed it into his parking space and turned it off then I tempted to start it again with no luck it turned over just fine the - lights came on brightly but just know start. I left for approximately one hour came back to the car and it started right up I pulled the negative cable on the battery and confirmed the alternator is not my problem. The car stayed running for about 15 or 20 minutes and didn't shut odd. I put the car in drive and any immediately shut off. Tried starting again with no luck. Someone told me that I could possibly be something to do with a built-in security system and to find out how to either turn it off or reset it with some kind of code with turning the key on and off etc. Is there a diagnostic tool that I can use to tell me what's is wrong or do you have any suggestions or advice for me? I just bought the car into an excellent condition I'm the fourth owner now and I have all of the paperwork from the lady that I bought it from with all of the work that's been done to it and not any parts that she has bought for the car. Any help would be greatly appreciated.