What SHX said. Only one dedicated circuit energizes the starter. But there are plenty of places for a fault to lie. Including (but less likely) a new battery!
The battery is low on the possible culprit list of course, since you just replaced it. So the list goes like this:
1. Ignition switch. Easy to check for voltage at the Red w/blue wire with the right probe. Of if that is not workable, you can remove the main connector from the back of the switch and test it with an ohm-meter.
2. The Red w/blue wire itself, between the ignition switch and firewall.
3. As mentioned, the neutral safety switch if you have the C4 automatic. If you have a manual transmission you still have the same connector behind the motor. Look for a 4-wire square connector with two Red w/blue wires and two Black w/red wires.
The body-side connector will have all four wires. The engine compartment side of the connector will have two Black w/red wires, but a simple loop of wire (as a jumper) between the Red w/blue wires.
This Red w/blue jumper is often at fault with no-start situations after all these years. They just get old.
4. The same Red w/blue wire should be on the "S" post of the starter relay/solenoid. It's usually on the left of the two small connectors, but depends on how they mounted the relay to begin with, so look for the "S" molded into the plastic.
The Brown "I" wire does not have anything to do with the starter cranking either. But without the Red and blue wire, nothing happens.
5. The starter relay itself. Old ones are pretty reliable, but they still get old and die. New ones are crap for the most part, right out of the box. So if this is a recent replacement I would suspect it.
Easy to test these too. If you jumper between the battery cable and the terminal where the Red w/blue wire is and the starter cranks, your relay is good and the Red w/blue wire is your suspect.
Just this week another member found that his was hacked in a few places and one of the splices had come apart.
Darn PO's!
Remember in all this testing that if your battery is connected, make sure the transmission is in Park if an auto, or neutral if a manual. Otherwise not-so-funny things can happen.
Good luck.
Paul