My 1978 Clara natural gas furnace got the annual "vacuum out, clean the flame light, oil the motor bearings, vacuum out all the vents for at least 10 feet in" it ritual a couple of weeks back. It hasn't gotten cold here yet, but slightly after it changes to regular operations cycles, I'll complete the process with its 1st new filter. The second one will be put in around Jan 1st.
The Clara is old school with no electronics other than the digital programmable thermostat I installed 11 years ago. The whole system works great and has never had any issues. Our gas bill (hot water heater included) is about $110 per month from November to February. Simple and effective.
Your natural gas supplier will generally check the heat exchanger for cracks for free. That's important on an older gas furnace. A cracked heat exchanger could be the source of carbon monoxide poisoning.
I always kept a spare thermocouple clipped to the furnace cover (so even I could find it). I ended up using it last spring when the home inspector found an unexpected problem with the furnace. The natural gas company guys will generally change it for you if it's required and you have a new part handy. One less trip for them and makes for a happy customer.
Nowadays I have a heat pump system. Not much to check - just replace or clean air filters and clean the outside unit's screen.