Check the 7.5 amp fuse AM1 that is under the dash to the left of the steering column and the 30 amp AM2 fuse that is under the hood. Don't just look at the fuse, use a meter to check the continuity of them.

Put the transmission shift lever in neutral and try to start the car.

 

 

Did you check the voltage of your battery? Just because it's a"new" battery, doesn't mean it's good. Also if you can have someone else crank the ignition while you read the voltage at the battery, does it stay at around 11.5 to 12 volts? Another thing it could be is the alternator.

 

This. Did you have any accessories plugged in or aftermarket electronics installed? Even with a new battery, parasitic drain can deplete the batteries in a matter of hours.

 

Not sure if you answered this question, but do the other parts of the car have power, such as the lights and radio?

Assuming the battery has a full charge, what I would check next is the start signal that comes from the ignition switch to the starter. Connect one lead from your voltmeter (or a test light) to the trigger wire (black color) on the starter solenoid and the other lead to ground. Then turn the ignition key and see if you get 12 volts when the key is turned. If you do, the starter is bad.

If not, more troubleshooting is needed.

 

 

Relays are a pretty reliable thing and 99% of the time are not the problem. As you found out.

Here is the sequence of events for the starter to start the engine:
Note that there are 2 distinctly separate things that must happen at the same time for the starter to work.

Process 1:
12 volts from the battery goes through the AM1 fuse to part A of the ignition switch.
Ignition switch contacts close when key is turned.
12 volts goes from ignition switch to the transmission park/neutral switch.
If transmission is in park or neutral, 12 volts goes to the starter relay energizing coil.
Starter relay energizes and closes internal contacts.

Process 2:
12 volts from the battery goes through the AM2 fuse to part B of the ignition switch.
Ignition switch contacts close when key is turned.
12 volts goes from ignition switch to the starter relay contacts.
If starter relay is energized, 12 volts goes to the starter solenoid.
Starter solenoid energizes and closes internal contacts.
12 volts from the battery goes to the starter motor.

You can take a wild guess and replace the starter, which has a high probability of being the problem. But rather than spend $175 to $200 on a guess, do the test I mentioned earlier.