I'm not Electrical Guru, but here is where I would start:
1. Take the wire (usually yellow) that goes from the ignition switch harness to the starter relay on the inner fenderwell. Put a test light or voltmeter there. Turn the key to "run" You should have 12 volts there. If you do then the key is telling the relay to send power to the starter. I would then check for power on your starter cable going to the starter.
2. I know that when I replaced my ignition harness, I had intermediate starting and it turns out that the connector from the ignition harness to the car harness was loose. So I just zip tied the connectors to hold it tight together. I'm sure Del will chime in.
