Car starting issue - help!

[Hartz]

I will start by admitting that I am a complete noob, and electrical systems are not even in the top 5 of things that I don't know. I throw myself at your mercy seeking help.

I have a Camaro with a TPI 350. I've been having issues with it not wanting to start after running for a while. Starter heat soaking, right? Well, no, I don't think so. Sometimes I could let it run for just a few minutes, and then it wouldn't start. If I let it sit for a bit, and fiddled with the key, eventually it would start. So anyway, this is all background.

Fast forward to this week when I'm trying to get her out of storage. battery is all charged up, but no start. There are no codes flashing, the radio, courtesy lights, all of that works. When you turn the key to run I can hear the fuel pump. Plus the fuel pressure gauge on the engine reads well. But when you turn the key to start - nothing. So a neighbor said it's probably the starter, and to determine that I could use one of those remote starting things. So I hook the wires up to the correct terminals on the starter, turn the key to run, and press the button on the little remote starter thingy. Sure enough - the car fires right up.

So is it the ignition switch that's gone bad? I'm assuming if the car would start with the remote starter and then run OK after that, that the coil and everything must be OK? The starter clearly worked. Should I look at replacing the ignition switch? Is there something else I should check first?

Any help appreciated!

Thanks!

[Debonair]

Is it a manual or auto trans?
If it is an auto trans, it could be the neutral start switch, if manual it could be the switch on the clutch.
If not one of these, you would have to see the wiring diagram, if it's available, to see what else is in the starter solenoid circuit.

[Hartz]

It's an auto - 700R4. I'll check the neutral switch. Thanks.