SIMPLE problem but you might have to get dirty. Need a multimeter or 12v test lamp, maybe an alligator clip jumper wire.
Circuit is simple Look at photo of starter relay with red arrow
The two small "push on" flag terminals are the relay coil. One gets power from the ignition switch when twisted to "start" the remaining terminal is grounded by the CENTER terminal of the neutral safety/ reverse light switch, which is mounted on the transmission
THIS IS AN AUTOMATIC, IS THAT RIGHT?
So disconnect the dark wire from the relay, the one which runs down the firewall, along the transmission. See if it starts. If not, the wire/ NSS switch has a problem. If so, the relay has a problem
From there, check the wire for shorts to ground. Disconnect the switch connector at the transmission, as well as the wire from the relay. That wire should be open. If not, it is pinched / shorted somewhere.
If this is not so, block the car safely, and put the transmission into ANY gear EXCEPT park or neutral. Check continuity of center pin of NSS to ground. Should be open. If not, either switch is bad, or something wrong with mounting/ internal rooster comb, use your judgement