Is it my starter or something worse?

For the past week when I try to start the car, at first I just get a click but on the second attempt it turns over. Yesterday when I tried to turn the car off the engine continued to run for a minute until it eventually shut down. Then ther weirdest thing is that the car tried to start itself, even with the key out of the ignition you could hear the car trying to start. All power eventually drained from the car. I had to jump it to start it and then it sounded like the starter war cranking in the background. This morning it was dead & wouldn't take a jump. Please let me know if you have any ideas about this. Thanks a million...