If you Mazda just goes click when you try to start it after driving for a while especially after using the AC or lights, there is a possible cause, easy test, and easy fix. Usually in this case, the car will sit overnight and then start fine or start fine after sitting for as little as 30 minutes to an hour.
It could be heat soak (increasing resistance in the main wiring when hot), starter solenoid, or the ignition switch but it is LIKELY that the battery is not getting charged when driving.
If the alternator is not giving a full charge while driving, you will slowly drain your battery and then when you shut off the car and try to start it again you would get the "click, click, click" when turning the key even if the lights work due to low voltage and amps.
If the alternator is good, the main cause of not fully charging the battery would be the belt is too loose and slipping some. Do not replace the alternator until you do this test.
Using whatever method you have, check the voltage of the battery. You can get an alternator and battery tester on the web for $4.99 with free shipping on Ebay. I have one and it works great. After driving for a fair amount of time with lights and AC on, shut down the engine. Now test the battery voltage again. You may find the battery has lost some charge.
Check the alternator belt to make sure it is not loose and is up to specs.
A loose alternator belt is the main cause of batteries not charging properly with no alternator light showing on the dash.
This may solve your problem as this is a common problem to most cars.
Don't waste money if you don't have to.
I own a Mazda, repair them, and everything I post I have done.

 

Mine is doing this. Will check the belt and voltage for sure now. THANKS!!