There really isn't that much to do on a gas furnace. Replace the filter, clean dust and rust out around the burner. If it is condensing type, make sure the drain is open. Probably a good idea to replace the igniters every 5 to 7 years.
+1
Gas furnaces burn very clean, so very little to really maintain. Here's what I suggest:
When you know it is working properly, notice and note the whole start up process, for reference. The exact process depends upon whether you have a standing pilot or some sort of igniter system. But typically after a "call for heat"
[edit - GravitySucks correctly pointed out I forgot that first, the draft inducer fan should start up, that should pull in a pressure switch a few seconds later, then....], if igniter, the igniters should start glowing red, after a delay to allow them to heat up the main gas should come on (you'll hear it) and it should take a few seconds for the main gas to ignite.
Note the flame and spread. After another 30 seconds or so, the blower should start. At that point,
the flame should not change - if the flame changes, that can indicate a cracked/rusted (rust might be an issue for you?) heat exchanger, blowing air into the combustion chamber, which could cause flame roll-out or some exhaust into the room.
I would make a point of watching ours start up at the start of the season, and whenever I thought about it - every few weeks I'd say.
Our new furnace is so sealed up, I'm not sure there's much I can even watch for. I need to check youtube for how to monitor these newer condensing units.
If yours has one of those glow igniters, a spare would be advised, maybe replace at this time as
travelover suggests. If it is a standing pilot, keep a spare thermocouple on hand.
-ERD50