Thanks for coming to the State, we all like tourist that visit, there are numerous places, that all of these post have mentioned and missed as well.
Also if you are in a no hurry kind of way when you get to Concord, NH get off of 93 and get on the old road Route 3 North, you will see everything, that you will miss traveling on 93.
If you are a hiker, and in good shape, and have the proper emergency backpack equipment, you can have a go at climbing the Presidential range, or if you like Tuckerman's Ravine is always beautiful.
The White Mountain National forest is a great place to drive through and picnic.
Look for signs near the Basin for Arathusa Falls not so great in the Summer / Fall, but dynamite in the Spring with the snow melt run off.
https://www.google.com/maps/@44.095599,-71.680813,3a,75y,90t/data=!3m5!1e2!3m3!1s9754287!2e1!3e10
Lost River, Santa's Village, Ruggles Mine, The tramway at Canon Mountain, Loon Mountain has Summer activities.
http://www.loonmtn.com/experience/summer-activities/summer-adventure-passesMount Sunapee has an artist fair and workshop, where juried artist show and sell their wares, glass work, jewelry, furniture.
http://www.mountsunapee.com/mtsunapeewinter/onthemountain/events/eventdetails/nhcrafts.asp