Fontainbleau

You don't need a tour guide. You just take the RER to Fountainebleau - Avon from Gare du Lyon. It takes about an hour and a half by train. There is a bus at the train station that goes to the chateau. The chateau is closed on Tuesdays. Entrance to the grounds is free and it has lovely grounds; there is a modest food service there but you can also take a picnic and eat in the park. We have never seen a line for admission to the chateau; your admission fee will include a self guide audio device. I think they do occasional guided tours but am not certain of that.

This is a lovely day trip and unlike Versailles which is tourist hell for crowds, Fountainebleau is never crowded.

Stayed in Fountainbleau 2 yrs ago and loved it, great sidewalk cafes and the palace is impressive, the gardens and the Fountainbleau Forrest are wonderful for hiking.

Actually the train ride from Gare de Lyon to Fontainebleau - Avon is only about 40 minutes. From there you can catch Bus #1 to the chateau. The entrance fee to the chateau is covered by the Museum Pass if you have one, if not it's €11.

Fontainebleau is not served by the RER but by trains that leave from the above-ground platforms at Gare de Lyon. Irritatingly, some trains are Transilien (Paris suburban rail) and some TER (national local rail) and so you need to check the two separate websites for both to see all the train times. But the same tickets are valid on all trains.

sorry to have some details wrong -- I forgot that it was not RER but regional trains -- but from Gare de Lyon (and FWIW these trains do not have available restrooms nor does the station in Avon ). It is an easy trip and the ground are lovely.

In 2010 I paid 15 Euro for going from Gare de Lyon to Fontainebleau. The bus to the Chateau stops in front of the tourist office opposite of the Chateau. It's ca 40 mins walk back to train station.

Another useful thing to note is that Parisian single-journey transport tickets can be used on the buses between the station and the town centre.

Viator offers convenient transportation, using ParisCityVision, to get from central Paris to Vaux-le-Vicomte (then 3 hours on your own) and Fontainebleau (3 hours on your own).
We did this a couple of months ago.
Viator/Paris City Vision provides transportation only, not a guided tour, which is what we wanted.
3 hours at each locations was enough time for us.
Seeing both chateaus in one day using public transportation is complicated and the Viator transportation made it easy to do both in one day and still be back in Paris for dinner.