Grafton Notch State Park is north of Bethel on Route 26, where the road cuts into a rugged section of the Mahoosuc Range. The Appalachian Trail passes through the park, and side stops lead to waterfalls, gorges, and views toward Old Speck Mountain.
This is not in Bethel village, but it is one of the main reasons travellers base themselves there. Start early if you plan a real hike; the short roadside stops are easier, but the landscape deserves more than a windshield look.