First 4-5 nights in Paris. Then, take 3-4 nights somewhere else and wrap up the trip with the last 1-2 nights back in Paris.
The question, though, is the "3-4 nights somewhere else" part. I had a few options...
1. Take fast train to Avignon, set up base camp. From there, check out places like Arles, Gordes, or, make it a wine focused trip and tour the Rhone valley (Lirac, Chat de Pape, etc.).
2. Take fast train to Avignon, set up base camp and from there, tour the south of France.
I was thinking of hitting Champagne, but only as a day trip out of Paris (same goes for Versailles). I understand that Burgundy is tougher to just stroll through - wineries are trypically more "by appointment only" and they are more spread out.
I am not opposed to renting a car. I figured I'd rent a car on option 1/2.