One of France’s sovereigns’ favorite retreat for over eight centuries, the Château de Fontainebleau has been the home of Saint Louis, François I, Henri IV, Louis XIII, Louis XIV, Louis XV, Louis XVI and his wife Marie Antoinette, Napoléon, Louis-Philippe, Napoléon III and his wife the empress Eugénie. It went through a lot of transformation over the centuries: Saint-Louis’ Middle Age castle left place to François I’s Renaissance château; Henri IV built new wings and a canal, Louis XV demolished some of the wings built by its predecessor Henri IV to replace them with bigger wings, he also had a pavillon (accessible by boat) built on l’Etang Aux Carpes; Napoléon replaced the western wing with a monumental gate (main entrance to the château) while Louis-Philippe had the aviary built under Henri IV torn down. Aside from furniture being looted during the French Revolution, the château de Fontainebleau was spared.
So, forget about Versailles! Leave the tourists to the tourists, the huge lines at the ticket booth to the Château de Versailles and take the T (transilien) train to Fontainebleau. The trains leave every half-hour from Gare-de-Lyon. A 6-zone metro ticket will take you there. You can ask for the combo train-bus ticket (16€, it includes the bus fair in Fontainebleau and saves you a few euros) or buy a regular return trip and walk your way to the château through its lovely park. The park is a great place for a picnic if you like a lot of trees, subdued light, quietude (priceless when living in Paris) and a few teenagers kissing in the grass in the distance. The bus to downtown Fontainbleau stops in front of the train station. If you’d rather walk to the château, which I highly recommend, take a left when leaving the train station, go up rue Charles Meunier, then rue Bernard Palissy onto rue Antoine Cléricy, which will take you to the entrance of the park. There walk along the Allée Napoléon all the way to the Grille du Bois d’Hiver. You have arrived in the gardens of the château. (click on the map icon below to see the path.)
Château de Fontainebleau 
Main entrance (Ticket window)
Cour des Adieux or Cour du Cheval Blanc
(ticket window closes 45 min before closing time)
Closed on Tuesday, January 1, May 1, December 25