Going to France for a week in September, seeing Paris and Palais Idéal. What in Lyon and beyond is also worth seeing?

My mom and I are planning out a trip to France in September. We already know we're seeing Paris and I always wanted to see Palais Idéal, so that will bring us out by Lyon. Once we're there, what else is worth seeing in Lyon and beyond? Any by beyond, that could be by car or by train, since we might need to rent a car to get to Palais Idéal, and the train line could take us to another major city.

Part of this is trying to decide if its worth flying into Paris and then back to NYC from another city, or just do a round trip and have to backtrack to Paris.

Anything to note in Lyon that we wouldn't get in Paris? Any cities connected to Lyon by train unique enough to be worth fitting in? Or would trying to fit another destination within a week be rushing it too much? Part of me is considering following the train line down to Barcelona, but even if we add on a few days for travel time, I worry that might split our time too much? We're trying to balance making a list of things to go see and do versus leaving room to take in the vibe or do something spontaneous.