One way to check in this situation is to get a remote starter switch at Autozone or some other parts store. It has two wires with alligator clips and a trigger style switch. One clip goes where the battery connects to the starter and one hooks onto the solenoid wire. Unhook the ignition wire from the solenoid. If your car will not start, try starting it at the starter with the remote switch. If it starts you have a problem somewhere inbetween. Most likley it will be your ignition switch. If you have a multimeter you can remove the solenoid wire and stick the red probe in the wire connector and ground the black probe. Have someone turn the key to the start positon and hold it and see if you are getting 12 volts to the wire. What you have described sounds to me like a bad igniton switch if it is worn or shorted you car could just stall out just as if you had turned the key to "OFF". Secondly I would suspect the neutral safety switch. At any rate that remote starter switch can keep you from getting stranded.