Starter engages with shift lever and when just turn key to on

Gremlins, that's what you have. You have something grounding at the wrong time, or when it shouldn't be. In nuetral and park, you have a solid ground going to a solenoid that let's you start the car. In any other gears you don't. In your case, your ground is continuing past this point. My guess here, with your brief explanation is that the key switch is bad.