Why does the furnace turn off after it reaches the correct temperature and then immediately turn back on? Is this a furnace or thermostat problem?

Could be either. If you bypass the thermostat and the problem continues, you can rule out the thermostat. Check furnace for a status LED. If present, is it blinking?

High heat loss in the structure can cause this as well as a improperly adjusted heat anticipator or in the case of a digital style thermostat improper programing.