My heater does not want to turn on, yet my thermostat tells me it is on and my house is very cold. What type of troubleshooting can I perform?

Start by switching the thermostat fan selector to on, see if the blower runs. This will check the power supply, etc.

Where is your furnace, is it a place that you can see what it is doing when you turn it on.

With more info I might be able to help you.