Also, is buying a one-day pass covering all transportation needed such as buses and trams the cheapest way, or is there any other cheaper way? Many thanks for answering!
From Chateau Rouge take Metro line 4 towards Marie de Montrouge until Chatelet. Transfer to Metro 14 towards Olympiades until Gare de Lyon. At Gare de Lyon take Transilien R train towards Laroche Migennes until Melun (or any Transilien train that stops at Melun, which you can see on the screens listing stops at the head of the train platforms, such as trains heading to Montargis and Montereau). At Melun station, take bus line 34 in direction Place Verdun until stop Chateau and it's a 3 minute walk from there.
Price for the Metro and train to Melun is 8.20€ and can be completed using a single RER ticket bought from a ticket window (ask to go to Melun and the ticket seller will provide the correct ticket & price. You may want to write the name Melun on a piece of paper to show to the ticket seller, it's pronunciation is opposite of how it looks "Moo Lunh"). The bus 34 to Chateau is 1.80€, requiring a separate Ticket t+. If you buy it directly from the bus driver it'll be 2€. You can buy a Ticket t+ from the Metro ticket seller or from ticket vending machines at Chateau Rouge (or any other station).
Here's a schedule for Transilien R trains to towards Fontainebleau for Monday-Friday.
I would buy single tickets for Paris to Melun/Fontainebleau unless you plan on repeating this trip multiple times during the week
For Paris itself, a one day pass (Ticket Mobilis) makes sense when you're taking more than 4 one way rides during a day.