HVAC  control circuits & troubleshooting.

The Thermostat gets  the "hot leg" of the 24 volt control voltage from the transformer which enters the R or red terminal on the Thermostat.

Upon a call for heat the switch between  Red and White closes, this sends the "hot leg" of 24 volts out on the White circuit which is the heat circuit.

Upon a call for cooling the switch between Red and Yellow closes, this sends the 24 volt "hot leg" to the AC unit outdoors on the Yellow circuit.
Note there is an Auto switch and a On switch for the fan circuit, the Green circuit.
In Auto the Green is energized whenever Yellow is via the closed switch in the Auto fan position, this operates the fans high/cooling speed as required by the thermostat.

Upon a call for Fan On, the switch between Red and Green closes, this will bring the fan on continuously, the heat speed is low speed and controlled by a time or temperature delay not the fan relay unless electric heat.

Some Thermostats will be parasitically powered by the furnaces transformers 24 volts, in this case the Thermostat will require the 24 volt Common leg of power as well as the Red hot leg of 24 v power, this will be to power the thermostat if not battery powered.

If you ever find no heat or cooling the 1st thing to do is try the fan switch to On and see if the fan comes on or not, if it does this proves you have both high voltage and  the 24 volt low voltage control power as well. This saves you having to check breakers and fuses etc.
If the fan will not run  you will need to check the breakers and time delay fuses in any service disconnects, as well as any 3 to 5 amp auto motive type fuses that the circuit board will have to protect the transformer in most cases, some older units may have no fused protection  or have an inline fuse that uses glass cylindrical type fuses.

If the breakers are good the powers being sent to the furnace then the transformer must be tested for 24 volts being read on its output with line voltage entering it.