Are you set on the Paris as a base for day trips strategy? I'm not trying to be overly critical here, I just think the Paris as base for day trips is not optimal. This strategy makes the most sense when there is a single train station that you are staying close to, when the hotels are not too expensive, and the day trip destinations not too far away.
You will pay a lot for a decent hotel in Paris and then incur significant extra cost for the train, etc. And the train station that you need to leave from to get to your day trip destination may be on the other side of town. So 1:15 can easily add up to 2:00 each way, metro changes, etc. Murphy's law, the metro you need is the deepest underground and the escalator's not working, etc. My view is some cities are more conducive to a base + day trip strategy and Paris is not especially one of them. (Madrid is a good city for this strategy). It's your trip though so if you are set on it then go for it. Rome2rio give you basic information and cost estimates, but you want to go to a local site for more complete/up to date information. You will want to get a guidebook and figure out the practicalities of where trains leave from for a specific destination, how close does the destination train station get you to where you really want to go, etc.
Versailles is an easy one and it's just on the RER so not far or expensive. It will be very, very crowded in June; don't go on Sunday when it is mobbed. Chartres is very worthy and do-able at 1:16 according to rome2rio, but check on the cost of the tickets. rome2rio is showing $20-30 one way. Reims is less than an hour but for two people round trip, plus a $140 hotel room in Paris--you get the idea. Fointainbleau could be workable as well, I was not blown away by this one but maybe it's because I was tired and had already been driving for several hours that day. The town itself is very pleasant. Giverny is another one that is closer to Paris, again could be expensive to get to--I think you can only take the train as far as Vernon and then a taxi from there. (It would be great in June though!)
Rouen in Normandy is worth considering, too.
Depending on how much time you have, a couple of nights in Reims with a day trip to Epernay could make sense.
You can probably get a bus tour for the Loire that would cover two or even three chateaux--you can cover a lot in a little time in the Loire. We were there for one full day plus a morning and saw four chateaux.
There is enough to keep you busy in Paris for four full days minimum. You can probably save booking specific trains well in advance.
For Provence, Avignon is well situated as a base for train journeys and the "base + day trips" is more workable there I think. A lot of the train journeys are pretty short. Although Arles is usually recommended as a place to stay over Avignon, I think the latter is the most convenient.
Note: you do not have to go all the way back to Paris to fly home, go open jaw if you can. Actually I would probably fly into Marseilles and out of Paris.