If you want to stick exclusively to public transport, then I’d look into Paris-Lyon-Arles, and include a day trip to Chartres/Rouen/Reims to round it out.
The Loire is pretty wonderful, but I feel like a car is needed to get the best of it. Also, you can get start to get tired of chateaux after seeing about 5 of them. You can TGV to Tours and rent a car there. Then stay in a smaller town.
Provence is also best done by car. Once you get away from Avignon/Arles useful transport is scarce, but you can get there via TGV. Provence covers a lot of sightseeing ground, and could easily fill your trip. You have the Luberon and its villages, Aix, Arles (I hated Avignon) and on the northern edge Orange and the Dentelles de Montmirail is prime wine country surrounded by world famous vineyards. It’s all gorgeous country, with great food, markets, and stunning Roman ruins.