Day to day operating cost can be expressed as dollars per therm (or 1000 BTUs). Both the gas and oil suppliers can tell you what their cost per therm is. That will tell you the least expensive fuel at this point in time to use. Gas may be natural pipeline gas or propane.
Generally, gas is cleaner, does not require truck delivery (except propane), will not leak into the ground, has generally less odor, and produces less pollution (soot, sulfur oxides, CO). Because gas furnaces are more common, efficiencies (heat delivered to your home from a therm of fuel) has gotten very high. Gas appliances frequently do not need chimneys, but use PVC pipes to exhaust moisture and combustion gas to the outside, and draw combustion air to the unit. If you are faced with replacement of your equipment (furnace, heat exchanger) and the cost of fuel is all the same, and you have a gas connection, I would be inclined to go with gas. Propane may require lease or purchase of a tank, and has a lower heating value, so the decision is a little closer.