1994 Other Jeep Models A Short in Some Circuit?

I have a 94 Jeep Grand Cherokee, 155,000 miles, that my wife and I purchased from a private seller a week ago. The first time we started it, it turned over and over but wouldnt start directly. We had cold weather, and it had obviously been sitting for a few days- I just figured it was cold. After it started, on the third or fourth attempt, it drove wonderfully and I had no "check engine" light or anything. The owner, who apparently bought the vehicle at an auction, said the start up problem was because the battery was too small. Since then, I have changed the battery and OEM terminals and still have the start up problem. Other electrical issues are: no radio (I changed and checked fuses, no rear driver turn signal ( I changed bulbs), sometimes dim lights wont work, sometimes temp guage stops working. Once, radio came on and stayed on AM530 w/ no volume and no controls. Once it came on while driving and again no way to control it. Once heater didnt work. These problems are all random, but the start up issue seems to be getting worse. It seems like a short somewhere in one of the circuits. I bought a Haynes manual, but I am unsure where/what to check for first. Can you help?