Which is the best way to save money on heat?

I used to think it saved money to lower the thermostat when I left the house but a former landlord told me that raises the cost of heating because the heater must then work harder to get to the higher temperature when it's turned back up. My furnace is natural gas so which is the correct way to save money, lowering the thermostat or keeping a constant temperature?

Heat loss is based on temperature differential. Long furnance cycles in general are more efficient than short cycles. For these reasons it is more efficient to turn down thermostat when leaving. An exception is when using a heat pump (cannot perform a large temperature swing, so turns on more expensive backup heat).

Check out an energy star thermostat at the hardware store. It is not one with a fixed temperature, it is one that turns down the temperature automatically when you are away.