1999 Saturn SL1 Starting my Saturn

I have moved from Michigan to South Carolina and this is my first summer with such hih temperatures - not sure if that's part of the problem. I was having trouble with my car starting a few weeks ago - I would drive somewhere, run inside and come back out and my car would not start. If I waited about 30 mintues, It would start. Sometimes if I hit on the starter it would start right away, other times I had to wait the full 30 mintues. Anyway, I had my starter replaced and it's been starting fine until today. I ran inside the store and when I came out, my car would not start. It was different this time though. Before, when I turned the key, battery power came on, the radio came on, etc. This time when I turn the key it fades in & out, the RPM revs up and down and shakes. When I turn the key completely over to start the car, all I get is a quiet, high-pitched beeping/squeal. I have no idea what is going on with my car and since I've recently moved, I don't know who to take my car to without paying an arm & a leg. Any help is much appreciated!