We converted from heating oil to natural gas two years ago. When we moved in the existing furnace was venting CO into the house, so I needed to do something. We were lucky since, even though there were no natural gas appliances in the house, the gas line went right to the home (I just needed to have the line tested and a meter installed, then add the lines inside.
I had a plumber do the gas line work, and I installed the furnace myself. It was not an easy job (cutting/fitting te ductwork was the worst of it).
Savings: We are saving a lot of money due to the conversion, mostly because I also scrapped our electric water heater and put in a gas one. The old oil furnace was approx 70% efficient, the new gas furnace is 93% (and the ones on the market now that qualify for the federal rebate are 95%).
The good news is that you've already got NG running into your house, so that will greatly reduce your expense (provided the service is adequate for running a furnace, which it probably is). Do you want to continue to use your radiators (i.e. do you want to install a natural gas boiler) or do you want to convert to a forced air heating system? It would be cheaper to stick with the water-based system, and I believe it might be possible to set things up to also provide your domestic hot water at the same time (which is a lot cheaper than heating water with electricity). If you are happy with the radiators, then there's no only real advantage to converting to forced air unless you also want to install central air conditioning at the same time.
Note that the high-efficiency furnaces and boilers take in air from the outside and vent to the outside through PVC pipes, so those ducts will need to be run, but that's usually not very hard to do.
Here's where I bought my furnace. There aren't many companies that sell HVAC equipment to DIYers, but these folks do.
Alpine Home Air Products: Contractor-grade heating, air conditioning and indoor-air quality products at wholesale prices.
Heating oil prices and propane prices are expected to be very high this winter, while NG prices are expected to rise far less. But, things could be different in the future.
Advantages of getting rid of the oil burner: Less annual maintenance for the equipment, no chance of an oil leak and al the environmental badness that goes with it, the fumes of heating oil are no longer permeating my garage, and I regained 10 sq feet of floor space in my garage where the tank used to be.
Unless you are handy and itching for a big project, this is something to contract out. It's got all the elements: electricity, sharp metal, explosive gases. Woo hoo!