Trains from the Gare de Lyon in Paris arrive once or twice an hour at Fontainebleau-Avon on what is now called the Transilien R line.
Currently (as of 2021) a full-price one-way point-to-point ticket from Paris to Fontainebleau-Avon costs € 8.85. This does not include the bus from the station to the palace, so it most cases it would be cheaper to get a one-day “Mobilis” pass for zones 1-5, which costs € 17.80 and includes transportation in Paris as well as the bus in Fontainebleau.
The train to Fontainebleau-Avon takes about 35 to 40 minutes, usually with stops on the way in Melun and Bois-le-Roi.
This photo shows the kind of trains they were using on the R-line in 2014, but I understand that since then a new generation of trains has been introduced.
The R-line, as it has been called since 2004, is one of eight lines comprising the Transilien network of suburban trains run by the state-owned railway system SNCF.
As I have mentioned elsewhere, the name Transilien was introduced in 1999 as a brand name for upgraded suburban trains running in the Île-de-France region on lines that have mostly been in operation since the 19th century. Unlike the five RER lines (Réseau Express Régional), the Transilien lines do not run through Paris in tunnels, but start or end at one of the traditional Paris terminals or at La Défense.
From the train station Fontainebleau-Avon it is easy to get to the palace by taking the local bus # 1 in the direction of “Les Lilas”.
If you have a transportation pass such as the Mobilis day pass for zones 1-5, then the bus ride is included in your pass. Otherwise the bus fare is 2 Euros (as of 2021).
These are local buses which are very popular with the local population. Between the station and the palace there are seven stops, with lots of people getting off and on. Your stop for the palace is the eighth stop, which logically enough is called “Château”. This is the stop after “Église Saint Louis”.
At the bus stop there is a side entrance to the palace grounds, so you don’t necessarily have to walk around to the front.
My photos in this post are from 2014. I revised the text in 2021.