Natalie, what a great trip! The short answer is that there is something wonderful in every direction, and you have to set priorities.
Yours are pretty clear, including the two most famous wine regions in France. Your itinerary would be simple but for Bordeaux, which is a bit far from your other stops.
Since Bordeaux brings you to the Southwest, I suggest you consider a stop of at least several days (or longer) near the Dordogne valley. Perigordaise cuisine is great, and there are many wonderful sights to see in this beautiful region.
One way to connect all these dots might be Paris - Loire - Burgundy - Lyon - Provence - Perigord - Bordeaux, with a flight to Italy (or maybe ending Bordeaux - Perigord, with a flight from Toulouse).
Alternatively, reversing this itinerary to travel roughly south-north avoids the chilliest weather; fly to Italy from Paris. (On that same score: Why not start in Italy? Italy ravishing in April; France gorgeous in May.)
Two comments on this plan, though.
Either itinerary can be realized by train or car, except for Perigord, but you will probably want to rent a car in Burgundy, Provence, Bordeaux, and the Loire (and Perigord).
Also, this would be an ambitious itinerary for three weeks, but as the start of a seven-week trip I think it is too much. Some sprinting is OK, but you need to vary the pace.
So my final suggestion is for you and yours to get a good map of France and a general knowledge of rail and driving travel times, then decide which destinations are least worth the time it will take to travel to them and brutally eliminate them. An extra day or two at you best destinations may be more rewarding than a long trip for a quick stop in an additional place.
Good luck with it! And very best wishes.