I recently had a furnace come in- complaint was that nothing happened- no fan, no heat.. nothing.
Whenever I have a furnace problem, the first step I take is to go to the thermostat. This one had a simple heat only, mechanical thermostat. I pull the cover off, switch my multimeter leads to read amperage and hook up the leads across the thermostat terminals.
On this furnace, I immediately read a 1 – 1.4 amp draw, which dropped quickly to around .2 amps. This told me the time delay fan relay in the furnace was good (on most furnaces built in the past 20 years, this will be a valid test- the exceptions will be new models with “fan control” circuit boards, or models where the board has been retrofitted with a Dinosaur brand fan control board).
After about 40 seconds, the blower came on, and the amperage rose a bit to around .5 amps. After another 45 seconds, the amperage rose again to about 1 amp.
What these readings told me was 1- the relay was good and the furnace had power (the initial high amperage reading was the heating element in the relay heating up), the second amperage jump (after the blower came up to speed) told me that the limit switch and sail switch were both OK, and power was getting through to the circuit board, the last amperage jump told me that the gas valve was getting power, and the circuit board was opening it. (A side point is that the final amperage reading- 1 amp- is the setting that the “anticipator” should be set at on mechanical thermostats).
For that job, I was lucky- the only problem was a bad connection in the thermostat- I repaired that connection and the furnace worked fine, but I hope you can use this method of troubleshooting to diagnose furnace problems (if you have a combination heat and AC thermostat, the same method can be used, but you have to find and break the correct thermostat wire to the furnace, which is usually easiest to find right at the furnace, and will nearly always be one of the two blue wires).