Fontainebleau has a good range of restaurants for a town its size, concentrated around the central market square (Place de la République) and the streets between the palace and the town centre. The emphasis is on traditional French cooking — brasseries, bistros, and a few more ambitious restaurants aimed at the Parisian weekend visitor.
The Saturday market on Place de la République is worth timing your visit around: local producers bring cheese, charcuterie, bread, and seasonal vegetables, and it's easy to put together a picnic to take into the forest.