Littleton itself is walkable — Main Street, the riverwalk, and the town centre are all within easy walking distance of each other. The town is small enough that a pair of comfortable shoes will take you everywhere you need to go in the centre.
A car is essential for anything beyond the town centre. Franconia Notch State Park, Sugar Hill, Bretton Woods, and the Kancamagus Highway are all driving destinations, and there is no meaningful public transport to reach them. If you're relying on the Concord Coach bus to arrive, plan to rent a car once you're here if you want to explore the region.
The White Mountains Visitors Bureau operates a seasonal shuttle in the Franconia Notch area, but coverage is limited and schedules change. Don't count on it as a primary way to get around.