A comprehensive rail network ensures that most towns in Ile-de-France can make comfortable day trips from Paris, but make sure you know the right station to head out from (Gare de Lyon for Fontainebleau, Gare St-Lazare for Vernon, Gare du Nord for Compiègne, or Gare Montparnasse for Chartres). RER (commuter) trains tunnel through central Paris en route to Versailles, St-Germain-en-Laye, and Disneyland.
A handful of venues need other means of access. To reach Giverny, rail it to Vernon, then take a taxi or local bus (or bike). To reach Vaux-le-Vicomte, head first for Melun, then take a taxi or local bus (in summer a shuttle service). Senlis can be reached by bus from Chantilly. And note that Fontainebleau station is in neighboring Avon, and getting to the château means a 10-minute bus ride.
The best way to crisscross the region without returning to the capital is by car. There's no shortage of expressways or fast highways. Expressways fan out from Paris in all directions: A13 northwest to Versailles and Giverny; A1 north to Senlis and Compiegne; A4 east to Disneyland; A6 southeast to Fontainebleau; and A10 southwest to Chartres.