Safety first

Test every carbon monoxide detectorPress the test button on each one; a furnace that burns fuel can produce CO, so this isn't optional.
Install a CO detector if you don't have onePut one near sleeping areas and on every level of the home.
Replace detector batteries nowFall is the easy-to-remember moment, before you're relying on them daily through winter.
Clear the space around the furnaceMove boxes, cleaning supplies, paint cans, and anything flammable at least a few feet back.

What you can check yourself

Replace or clean the filterA clogged one restricts airflow and makes the furnace work harder.
Open and clear all vents and return grillesMove furniture, rugs, or curtains blocking them.
Check the outdoor exhaust vent from the outside only if gas furnaces with exterior ventLook for leaves, nests, or debris blocking it.
Flag any rust, damage, or a disconnected vent section for a technicianDon't attempt to fix venting yourself.
Swap thermostat batteries and switch it to heatSet it a few degrees above room temp to confirm the furnace actually kicks on.
Recheck your thermostat scheduleMake sure it still fits your household's routine before you're relying on it daily.
Book a chimney/flue sweep and inspection if you also run a wood-burning fireplace or stove if wood burning fireplace stove or insertNeeded yearly regardless of how often you use it.

Run the furnace for several minutes before you need itDon't wait for the first cold night to find out it won't start.
Note any strong odors or loud rattlingA slight dusty smell on first startup is normal, but anything beyond that needs a professional look.

Schedule your annual furnace tune-up now, not in NovemberTechnicians get booked solid once the cold hits, and this is also when they catch issues like burner or ignition problems before you're depending on the system every day.
Let the technician handle the burner, ignition, and gas/electrical componentsThese aren't homeowner tasks.
Ask about the condensate drain if you have a high-efficiency furnace if condensing furnaceA blocked line can back up and cause water damage.