The car won't start; lights, ac, radio etc come on but car won't turn over or even click; checked battery cables; put different battery in; still won't start

With a voltage meter see how many volts are at the starter terminal. If not 12+ volts you have a bad connection. Check connection at the starter solenoid. Does it have 12+volts? Short the solenoid. Does it put voltage to the starter?

Even though all your other electrical bits are working it could be that your battery doesn't have enough juice to crank the engine. It doesn't take much power to run the radio, lights, etc, but cranking the engine requires the battery to have a very strong charge. If your battery is more than 5 years old, that's very likely the issue. I'd start there. Also, be sure to check the battery terminals and ground for corrosion and clean if necessary. If replacing the battery doesn't solve the problem it could be the starter solenoid or a loose electrical connection somewhere.

start by following your negative wire off of the battery it should go to metal on the engine or to the frame (a fender bolt is ok) try putting your negative jumper cable from battery to any metal on engine. turn on headlights wiggle cable to make sure you have good connection, see if head lamps come on. If lights come on replace ground cable from battery. if not remove jumper cable and start looking at positive cable.check it for a good connection to battery and also to cable . if connections are good there follow the positive cable down to the starter solonide.you should check your battery voltage at this connection. If its good and the connection is good but you still have no power replace starter solonid.. I suspect the positive cable going to the starter will be bad or the connection will be bad.

Some vehicles have a 50 to 60 amp main power fuse would check for that.

You need to charge your battery up more. Either get a battery charger or leave the jumper cables hooked up for a longer period of time.
Your battery was probably completely dead. You might even need a new battery...they really don't like to be completely drained.

check fuseable link near battery if it's blown would pay to look for short circuit before doing much more

more then likely its a bad starter. or the cable to starter.

Did this occur straight after changing out the battery, or after sitting with the new battery installed?

Have you checked the new batteries voltage and voltage under load, eg. (Voltage when headlights are on) could be a flat new battery.

Or could be load on the battery after key is turned off, something draining the battery overnight like a boot or interior light, radio left on ect.

Unlikely a short, or you would be in a world of burnt wires from the battery cables.

clean the battery cable terminals and battery posts till it show's bright metal , put some grease on the posts and tighten the terminals.

look under hood remove cover for relays and fuses , should be a main fuse link where pos cable from battery and a ground come together with a fusible link in between it might need replacing