We will be in Paris for a week, arriving Saturday morning through the next Saturday morning. We will need unlimited travel within Paris and to the World Cup stadium, and to come and go some days to Fontainebleau (same day). What is the best pass option for us, with convenience being more important than cost if a trade off has has to be made. Also, could you explain if the RER train from Gare De Lyon differs from the other train options and how.
Arriving on Saturday and if you plan on using the trains a lot, then buy a 2-day 1-5 zone Paris Visite ticket to cover you for Saturday/Sunday. Also, at the same time, buy a Navigo week pass which will start validity period on Monday and cover you through the rest of your time in Paris area. Both will cover you for everything (Metro, RER, Transilien, out to Fontainebleau, the Stadium).
There are RER D trains from Gare de Lyon to Stade de France that would be your most direct train route between the two stations.
For Gare de Lyon to Fontainebleau by train you would take a Transilien branded train which leaves from the same ground-level platforms as intercity trains known as "Grandes Lignes" trains, not from underground stations for RER trains or Metro trains.