France is pretty large so there is no one place you could stay and travel out from to see all the different areas that France has to offer.
As an example I spent a month doing basically a road trip around France by a combo of rental cars and trains. I was able to get to Provence, Normandy, Brittany, Loire Valley, Riviera (Nice area), the Rhone-Alpes and Alsace.
I also spent a month in a rental apartment in Paris, seeing that city in depth and doing day trips by train to several near by cities including Versailles, Chartres, Giverny.
Comparing the two months I will say the the month in Paris was more relaxing, with so much time there was no pressure to see this sight or that sight on a particular day so it was a very loose schedule of sightseeing, and I spent a lot of time just enjoying the ambiance of the different neighborhoods and playing it by ear. On the other hand, I have to say that the month I spent 'on the road' was the most rewarding vacation I have ever taken. Even though it involved a lot of scheduling and timing of transportation and reserving hotels every few days, I was able to see most everything in France that I wanted to see.
It all really depends on what you want to see and where in France you want to go and how you like to travel (your travel style).