I'm not from New Hampshire but as it is my second favorite state (behind my own, of course), I will give a go at explaining it. I also have a personal empathy for NH, as it voted for Buchannan during my so-con phase and McCain during my moderate phase.
The state's motto - "Live Free or Die" explains it all. New Hampshire is probably the most ideologically libertarian state in the union and thus has an overall preference for Republican candidates. It has no state income tax and not many other taxes for that matter.
It also is probably the most (small-d) democratic state with a 400-member legislature and an great tradition of citizen civic involvement. That's why we trust it to select our Presidential nominees
It has made some 'wierd' choices though. It has a strong contingent of populists who sympathize with social conservatives - hence the election of Senator Bob Smith and the '96 primary victory of Pat Buchannan. I think it has less to do with an agreement on social conservative issues, and more to do with the idea of challenging an establishment. Buchannan's slogan was "peasants with pitchforks" and that kind of thing plays well in NH better than it would in any other state. 'Insurgent' candidates (Dean, Bradley, McCain, Buchannan, etc.) usually have a better shot in NH because of this populism.
NH is certainly different from Massachusetts and western Vermont but not all that different from Maine or eastern Vermont. (four of VT's easternmost counties went for Bush in '00)
And as for inclination, keep in mind that all three states you mentioned (Vermont, Mass, and RI) have GOP governors
