For 3 people what is the best sort of car for getting around the island of Naxos?

If one wants to visit the lesser frequented beaches away from the main town is an off road type of car necessary?

Most of the beaches should be accessible by a normal rental car. A Jimmy small Suzuki jeep is not a whole lot more expensive and gives you that extra ground clearance.
Here are some pictures of a recent mountain village day trip we took in a rental car.
http://travel.webshots.com/album/577950188puzxSj

In agree with stanbr. You can get the simplest car offered. Usually they are small, but you can get around Naxos very easily. Don't hesitate to ask from the car rental company to show you the different categories of cars, so that you are sure that you get the one that suits you best.
Note that usually in Greek car rentals you won't find automatic cars.

That's true, Adrian, a few years back there were only 2-3 automatic-shift rental cars available on all of Naxos .. but now there are more, time marches on, and more North Americans (who are stick-shift-challenged) have discovered Naxos, alas (you can partly blame stanbr & me for ongoing raves on various forums). However, best to inquire the day before, not day of, because supply is not unlimited. Also, to get an automatic, u will probably have to take a larger, thus a more pricey car ... such as a Hyundai Accent/Honda Accord size vehicle.

I have no problem with the stick-shift but it is a funny expression because the problem is really with the third pedal and the interaction with the accelerator rather than the stick itself!

That 3rd pedal is called the clutch. See under "Cars -- old fashioned vocabulary".

I think the car we hired in Naxos this May was either a group "A" or "B" and fine for all but the most rutted gravel roads. We drove pretty much all over the island and most of the roads are fine and relatively quiet. You can see my report, if you are interested, just click on my name. Naxos is a wonderful island to explore, hope you have a great trip.