There are many local walking trails that take you through beautiful forests, along the ocean, up little lookouts, and over streams or ponds. Morse Mountain provides a beautiful view with lovely forests, marshland, and beaches. You can even see to Mount Washington on a nice day!
If you are looking for a more rigorous day and do not mind traveling further, one of Claire and Christian's favorite hikes is the Bigelow Range in Carabasset Valley, and Camden Hills State Park also provides some lovely hiking and is close enough for a day-trip.