I assume they're going to be living in the Southeast part of the state--we lived in the Southwest part--DH actually worked in Vermont. My MIL lived in Manchester for 20 years or so, so we visited often--my DD also ran cross-country there sometimes. When we headed east to see MIL, it would be woods, woods, woods, woods, Dunks, woods, woods, woods...
The southern part of the state gets, on average, 5 feet of snow each winter. We typically had a week where it would get down to -10 degrees at night. But, people know how to dress, how to drive, how to shovel, etc. Summer is beautiful, especially if you enjoy the outdoors--many lakes, as well as that bit of ocean. We had a small family ski resort, less than 15 minutes from our house--DH volunteered as ski patrol, which got our kids free passes, rentals, and lessons. The elementary school had "ski club" every Friday--the principal and many teachers would join the families on the slopes.
No sales or income tax, but tourist stuff--hotels and restaurants--are taxed, and property taxes are high.
The state takes its politics very seriously--either side of the aisle. NH holds the first in the nation primary, and they treat that as a mandate to evaluate every single candidate. It's great from a civics point of view, but man--did I get sick of the polling phone calls! I still get them, since my cell phone has a NH area code.
You'll find the people to be reserved, but friendly, and they don't tend to say nice things about people from Vermont or Massachusetts. It's more of a friendly rivalry, though--our Target would be thick with Vermont license plates on weekends.