Live Free or Die is a pretty dramatic state motto. New Hampshire people are generally a little more reserved than that, but they’re a bit rugged and outdoorsy, tolerant of weather extremes, and somewhat into meat and hunting. Beer drinking and independent thinking are the norm.
You can visit various Shaker communities, such as the 1792-founded Canterbury Shaker Village, which shows the daily lives, environment, and crafts of the community. There is a great restaurant there featuring traditional dishes made with heirloom vegetables. Keene State College has an impressive art gallery focused on New Hampshire artists and others who have regional ties. If you want a serious educational/cultural experience, head to New Hampshire’s entry in the Ivy League, Dartmouth College. You can experience high tea at the Sanborn House Library at 4 PM every weekday…and some wonder why this area is called New England? Mt. Monadnock, near Jaffrey, is quite a peak. An ascent to the top and hike back down to the visitor center takes about 3┬¢ hours. Walk around the Squam Lakes Natural Science Center outside of Plymouth, where you can view bobcats, fishers (a kind of marten), mountain lions, great horned owls, and even bald eagles. Take a nature cruise there if so inclined.
Cider doughnuts and seafood are good New Hampshire foods. You can have particularly fresh, local foods like pancakes made from home-ground flour with homemade maple syrup near Franconia, pizza with organic vegetables and nitrate-free meats in North Conway, vegetarian Vietnamese rice noodle soups in Nashua, or a vegan buffet and salad bar run by 7th Day Adventists in Keene. Try home-style vegan American and Indian cooking in Concord that emphasizes locally grown, organic ingredients, or raw desserts or anadama bread made in Exeter. Some breweries have a decent number of healthy, vegetable-focused dishes, and vegetarians can also find good options at Thai, Vietnamese, and Mexican restaurants. At one interesting place in Portsmouth, you can have an espresso, one of several exotic coffee blends, or some tea as you peruse Turkish rugs.