Starting problems

I have 95 3.1L Grand Prix that does not always start. Sometimes it starts with no problems and other times I will hear a click but the car will not turn over. If I keep trying to start the car it does eventually turn over, I just tried changing my starter but the new starter did not solve my problem. What could possible be the problem? This has been going on for probably 2 years now. I have also got the car in the past by playing with the shifter. I am thinking neutral safety switch, but before I change another unneeded part I would like some suggestions. I am also not sure how to change the neutral safety switch.

You can test the neutral safety switch by checking for voltage at the purple wire attached to the "s" terminal on the starter solenoid. If there is no voltage when the key is turned to start, try shifting into neutral (instead of the normal park). If there is power at the "s" terminal in neutral, the neutral safety switch is bad. IF there is still no power at the "s" terminal, it could be a bad neutral safety switch or bad/out of adjustment ignition switch. Try moving the tilt steering column to different positions. If it starts then, the ignition switch on the coulmn is probably just out of adjustment. If there is power at the "s" terminal when the car won't start, then the issue could be a weak battery, bad/corroded battery cable, or a bad new starter.