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Yes, you have a fuel company come fill the tank with fuel oil. You will also need to have the furnace serviced every year which includes having the oil nozzle and oil filter (if it has one) replaced. The fuel tank is indoors or outdoors? You tend to have more trouble with them outdoors because there is a greater possibility of water getting in the tank, and then possibly freezing up the fuel line when it's really cold. Unless fuel oil costs there are significantly lower than they are here, fuel oil is going to be quite a bit more expensive per BTU than the natural gas you have now; if the house you're looking at has a much smaller furnace then you might be okay, but I would really look at how much the current heating bills are.
A central air conditioner just cools air, so you use it only in the summer. An air source heat pump does both cooling and heating (down to around 15-20ºF for most of them) and would be quite a bit more efficient than a fuel oil furnace for heating, so you could save on some fuel bills if you went that route.