How can I get my downstairs gas furnace to kick in quicker when I increase degrees on the thermostat?

When I wake up in the morning the temperature in the house may show 69 degrees on the thermostat. I increase the thermostat to 74 degrees to get some heat. It never comes on. I switch the fan from auto to on and then back and it may come on. I take the door off the furnace, put it back, I hear a click, then it kicks in. The colder it gets the longer it takes to come on, sometimes as long as 15 minutes. No problem upstairs, change the thermostat the heat comes on instantly. My house has high ceilings.How can I get my downstairs gas furnace to kick in quicker when I increase degrees on the thermostat?
adjust the anticipator.How can I get my downstairs gas furnace to kick in quicker when I increase degrees on the thermostat?
check the location of your thermostat if it located in a warm spot allready it may need to be moved or it is malfunctioning and needs to be replaced
the blower shouldn't come on until the furnace heats up other wise it will be blowing cold air, so there should be a delay between raising the temperature and the blower coming on however if the furnace is warmed up and the blower is not on then the limit switch might need adjusted or replaced.
I would suggest getting a programmable digital thermostat to put on your system; that way, you can program the thermostat to turn your system on just before you usually get up. Most gas-fired systems have to warm up before coming on, so that's probably the delay you're experiencing; it has to heat the air that's already in the system before the blower comes on, or you would have COLD air coming out of the vents. Your upstairs is almost certainly warmer than your downstairs most of the time, so that may account for the differences. If you still think there's a problem, then certainly it would be a good idea to have the system checked by a reputable repair service.
Your relay maybe sticking have it replaced
Are the burners burning clean and the full length and evenly ?

They should be a blue flame and not much yellow if any at all.
It sounds like you need a new thermostat