We recently stayed in Randolph, and were about 5 minutes away from the Appalachia Trail Heads...Fabulous, but not for the faint of heart or out of shape. Moderate in NH is difficult in most other places! We hiked the Valleyway to Madison AMC Hut, which is basically a rock scramble the last quarter mile plus, hut is awesome. From there you can go up to Mt. Madison, Adams or hook up with several other trails.
We also did Maple Way, Sylvan Way, Kelton, Dome Rock and Brookbank Trails. All excellent and at times include a lot of climbing and going through creeks. Fabulous views from Kelton Crag and Dome Rock.
If you own or manage Appalachian Trail, register now for free tools to enhance your listing, attract new reviews, and respond to reviewers.