Lots of great hiking within minutes of downtown! Franconia notch is not to be missed. Hike Indian Head (trailhead on Rte. 3 right before the Indian Head resort) or the Lincoln Woods trail (up 112 past Loon on the left).
There are more rigerous places to go around here, but anywhere off the main roads you can climb down beneath the trees and see this cool river that goes all around the mountain roads.