Buy a This car climbed Mount Wash sticker. You don't necessarily have to drive the hill though.
I'll second the Clark's recommendation-seems to be universally enjoyed. Plenty of places to eat afterwards in Woodstock between Woodstock Inn, Pemi Public House, Merlands, and Twin Barns. L_JE gave a thumbs up to Pemi Public House during his last ice climbing thread. I've never been
All my spots to eat are on the other side of the state in Littleton/Lancaster and down in Holderness so I'm no help there.
Littleton, I like Schillings pizza(owned by someone who was running as a R congressman). Freehouse is also good. For gas station pizza next to the trailer park, Simon's Market is good and is owned by an R state rep. The Inkwell is a good breakfast spot and is owned by conservatives.
Lancaster, I love Copper Pig's beer although some people on reddit have run into some flat beer. Good lunch spot IMO. Their menu may seem kind of bland, but check their FB page for specials which are widely varied.
If you decide to hit the Lakes Region, rent a boat on Squam Lake. Go to Walter's Basin for food. Golden Pond Country Store for gas station pizza and some of the cheapest gas in the state.
Avoid the hippies if coming in early July.