I've found that although our programmable thermostat is set at 68 degrees the heat only seems to come on when the house gets to 62 degress and then shuts off when it gets to 66 degrees. Is this a problem with the thermostat or the heater? We have just moved into this house and the heater is new but I am not sure how old the thermostat is.

The thermostat controls the furnace. Naturall if the furnace works at all it means the furnace is operation but the thermostat has lost its setting. Try a new stat or find a operating manual for the tstat and see if it is set up correctly.