Looking at the Fubicy site it sems that it is not as simple as one would think it would be in a country that loves cycling.
You're probably ok on regional trains (TER) in certain areas at certain times unless you're not! These presumably only allow you to travel relatively short distances.
For longer journeys, the grandes lignes(SNCF) , the majority do not accept bikes and those that do - you can't pre-book. It is incumbent on you to find the inspector and pay 10 euros for each part of the journey and a fine of 8 euros for not pre-booking!
It's possible that these rules are ignored but it's difficult to plan and makes me wonder if I should take the car or the bike bus. I'm trying not to take another air flight this year.