My AC did not turn on when the hot weather hit this summer. So I checked to see if the heater and fan would work, and nothing happens. The thermostat clicks as though it is trying to send the signal to the unit to either heat or cool, but nothing works. I tried changing the thermostat but that yielded no results. Could it be some sort of relay or the fan? All worked fine about six weeks ago.

Have you checked the breaker/fuse to the furnace? The low voltage power that starts both units start there. If all is OK check for power input to the transformer and then the output ,should be 24-28 volts coming out of transformer,

Let us know what you find