For central A/C, you'll need ducts and vents. Once you have the ducting in place anyway, you might as well use it for both your heating and cooling.
For a three floor house, you definitely want some kind of zoned system, with a thermostat on each floor. (I wish my 2-floor house was zoned.) You may want multiple cooling systems, or maybe one system with motorized dampers to control airflow to the various floors. Your HVAC contractor should be able to make a reccomendation of what will work best.
Whatever you do, do not skimp on the ductwork. Your HVAC contractor should run calculations on how big the ducts need to be. For many home builders, HVAC is a minor concern in the bulding plan. Make sure everything is sized correctly. Changing ductwork after the fact is very difficult. And if you don't have proper ducting, it won't matter how good your central heating using is.
For new construction, Geothermal is usually a good candidate. It can provide heating, cooling, and hot water with crazy efficiency. The installation cost is MUCH higher though. (There are tax credits that will offset some of the cost.) If you aren't up for geothermal, I would go for a traditional air-sourced heat pump for your heating and cooling needs. They are pretty efficient (at least in moderate climates). You are pretty much in the perfect lattitude for a heat pump.
However, I would not have the heat pump (air-sourced or geothermal) as your only heating system. Some kind of alternate (non-ducted) heat source would be a good idea. Heated floors, gas fireplace, etc. Air-sourced heat pumps lose their efficiency in very cold weather. And it can also be nice to have gentle air convection, vs a duct that is blowing in your face. You also have a backup system in case your central heat fails.
That said, ducted air is nice, because some regular circulation of air in all the rooms of the house avoids that "stagnant air" smell, as well as cutting down on dust, dander, etc.
Also, central air systems give you options for humidification and de-humification of the whole house. Look into that. It is usually a minor cost increase in the system, but can make a major difference in comfort.
Best Answer
With no duct work already existing, there's nothing really to "retrofit". What you'll be looking at is a new install in an existing house, something that's ABSOLUTELY doable. In fact, a few years back I got an estimate to install a new HVAC system and run a new trunk up to the attic and the estimate was about 12k.
The only thing that wouldn't have been new in that case was the gas line into the basement.
Newer high efficiency furnaces can be had which don't need to be vented out the chimney but can simply be vented outside the basement wall and partway up the side of the house. Likewise running propane or natural gas is pretty straight forward.
The most complicated part will be running the ducting. The ductwork should be sized out by a pro - size matters, particularly for the air return, which if too small can burn out the furnace.