Car won't start while battery is new

Are you saying the starter won't even crank? Did you try cranking in neutral? Check the starter wiring circuit.

You need to have the starter tested. Some of the national brand autostores will test it for you, if you take it to them.

The purple wire at the starter solenoid, should be hot/voltage with the key in the crank position. If it isn't hot, backtrack the circuit. Some possibilities, ignition switch--neutral switch--wiring circuit. Any testing at the starter, make sure the tranny is in park or neutral and a functional e-brake is set.

Forgot the diagram.

You could also have a fusible link at the starter gone bad and not allowing any power. Especilly if your lights work and nothing else does. They are easy to find they are smaller wires on larger ones.