First i would say this board heat a lot seeing all the burned spots on the pcb
I would add more ventilation with holes in the casing
Capacitors should be serviced or checked .... mostly the small ones around the ic
Check the parts solders for cold joints, cracked solders
It will fail sooner or later
The coil seem's to be the igniter IE high voltage pulse outputs ??
And the question about monitoring this device :
Without a schematic, i would not even try any external device additions or modifications or sensors ... i would need to probe many spots, one misshap and you could blow a part or create some problems
Mostly due to ac lines going into it too...
and finally your sentence: I am not too good electronics troubleshooting
Get or find a spare in case of ...
Furnace brand / model, and the pcb green side markings you have some who may tell the model ending with 37F

Seems an old Hitachi logo ??