Fontainebleau is easy to reach from Paris. The Transilien R line runs from Gare de Lyon to Fontainebleau-Avon station roughly every 30 minutes; the journey takes about 40 minutes. From the station, a short bus ride or a 20-minute walk brings you to the palace entrance. The town centre is walkable once you arrive.
By car, the A6 motorway runs south from Paris and takes about an hour in light traffic. On weekends the motorway can back up significantly; the train is usually faster and less stressful. From the palace, the forest trails and boulder areas are accessible on foot or by bicycle — bikes can be rented in town.