The thermostat should be powered by the 12v batteries. Check all fuses or circuit breakers that might power it. Then take the thermostat cover off and check for 12v there and also the ground connection. Using a couple of jumper wires from a known good 12v source you could try connecting the thermostat directly to check it's function and ability to run the A/C and furnace.
You might also do a Google search for the make and model of thermostat you have, look for an owner's or service manual to troubleshoot the thermostat.