We've only run the AC twice this year, went to turn it on Saturday with the latest heat wave and it didn't cool. I've cleaned the coils/fins on the outside unit, but have done nothing inside as of yet. However, the compressor is not coming on. We have dogs, and have to pay close attention to our filter, but don't always, so it is possible there is dirt inside. If I clean the evaporator coils might that fix whatever is stopping the compressor from running? Is there anything else I can check before calling a tech? What else could be wrong besides needing a new compressor (ouch)?

Sounds like an electrical problem, not a coil problem.

When a unit is not running the first thing you always is make sure the thermostat is calling for cooling and is in the right setting. Next is to check the power to the unit, high voltage and also low voltage. If your air handler is running that should mean that your transformer is OK, if you have someone to help you have them turn the thermostat up and down as you listen out by the condenser. If you have a loud click that means that your contactor is pulling in but does not mean that power is going through the points. Check the high voltage on the incoming power lines, if you do not have power then check your fuses or your breakers or both.