Make sure that the circuit breaker is on and not tripped...some of them, it's a bit hard to tell, so you might want to turn it off and then back on. There should be a disconnect switch near the unit...make sure that it is on and not tripped as well.
The click you hear...is it in the thermostat or a click at the compressor unit?
It depends on how comfortable you are with using a multimeter, as to whether it would be safe to check further.