How much public transportation is available in NH? What kind of public transportation is available?
Outside the cities, not much.
A lot. Less so in the Seacoast and especially the southern Seacoast (and the Salem area too according to the below information.) More so in the mountains, especially the White Mountains.
Be aware, though, that the snow the seacoast does get, tends to be wet snow, which is much heavier and more of an impediment to driving.
A typical seacoast storm might shut down travel for most folks, whereas a typical storm up north (which would be heavier) might not, because dry snow is less slick. For example, in the largest blizzard of this past winter, I was towing a trailer down from Grafton to Manchester, and drove back. I wouldn't have tried that same drive along the coast, even if the snowfall was only half as heavy.
Also, pay attention to the relationship between total snowfall, and days of snow. Certain areas will have more days where it is snowing, but less total snowfall, indicating a slower rate of snowfall. Generally-speaking, colder areas get more snow, but have fewer snowy days (it generally only snows when it is relatively warm).