Best way to get to Chateau Fountainbleau from SNCF train station?

There is a city bus, line 1, between the station and the château. It takes the same tickets as the Paris metro or buses.
And it is also realistic and pleasant to walk either to or from the station, going through the park: the "Porte Blanche" entrance to the park is about half a mile away from the station, then it is 1.5 miles through the park to the chateau, preferably along the canal.

What Balso said.

The Chateau at Fontainebleau is wayyyyy closer to the train station than its counterpart Vaux-le-Vicomte !

We took Ubers both ways. Nice and easy.

We did have a nice lunch at Antica Trattoria if you’re interested in a pizza stop.

No need for a taxi to reach the Chateau from the train station. That bus line suggested above is the best way.

I've taken the bus going there and the bus returning. Other times after the visit to the Chateau Fontainebleau, I walked back to the train station after having had lunch near the Chateau. Walking back takes about 40 mins.

The train from Paris to Fontainebleau will stop at Melun too.

See the "Cour des Adieux, " ie, the courtyard of the Chateau. In 2014 the reenactment (bravo !) of Napoleon's Farewell to the Old Guard was done in that very area where it took place in April of 1814.

Thanks, Everyone! We will probably walk at least one way dependent on the weather, of course. We're going with three adult children (well, two and a spouse) who are quite fit, so I'm not sure how well this out of shape 60+ year old would do with walking both ways!
Looking forward to it. Love all the tips available here on the forum. Thanks again!

Sorry for the mistake, Kerouac2! I do know the proper way to spell it, but was apparently in too much of a hurry to care. Lesson learned!