Verify the furnace has power. Check the breaker box to ensure that the furnace hasn't shut the breaker off. If the breaker gets overloaded, it shuts off to avoid a fire. Look for lights on the furnace when it is turned on.
Turn off the breaker to the furnace once you've verified the furnace has been getting electricity. This prevents the risk of electrical accident during the troubleshooting process.
Inspect the furnace wiring to look for loose or blackened wiring. Loose wiring in electric furnaces causes the electricity to arc, which builds up excessive heat on the wire. Eventually, the wire burns out. Any loose or blackened wiring requires replacement.
Check the furnace fuses. If one is blackened, the fuse likely blew out and requires replacement. Repeated blown fuses indicate an electrical short in the system that requires professional repair. Test the fuses individually by pulling them out one at a time, with the power off, and hooking each up with a multimeter. Touch the probes to both ends of the fuse to check for a zero reading in ohms. No reading at all indicates a bad fuse.
Clean the furnace and replace the filters to help eliminate buildup that causes overheating. Follow the instructions for your specific model to ensure a proper job. Call for professional service if the furnace still doesn't work after you restore the power.