I am planning to go to London, Amsterdam, and Paris sometime this summer for about two weeks. I would also love to travel in another section of France (besides possible day trips to Giverny or Versailles), but it seems like most areas are too spread apart to travel in anything but a car. Is train or bus travel (or even hiking or cycling) possible--or recommended--in areas such as the Provence or Normandy? I will not be able to rent a car, so I'm trying to figure out an itinerary for France.
Good news: You can travel extensively in France without ever renting a car. Like many western European countries, France has an excellent rail system. Here's the website that proves itFrance Rail Map The high speed train lines you see on the map will get you to your destinations at speeds of 150 to 200 mph!And here's another website: put in a starting and destination city and you'll see how good the trains are in France:Europe Rail ScheduleOf course it's true in France, as well as other countries, that trains can't go everywhere, so if you just gotta explore rural areas of Provence or Normandy, you'll may need the efficiency and convenience of a rental car for those sidetrips.
As Kent says, train service in France is excellent. While there is no long distance bus service within France, there is regional bus service which will get you to many places where trains don't go, including parts of Provence. However, you'll need a car to get to some of the smaller villages in Provence. There is also bus service in Normandy. but again not to every village.
If you read this page you will see that train travel is better and highly recommended by experienced independent travelers. Usually more economical than auto rental for one or two people.
Second choice is usually bus travel. It is slower but cheaper.
As others have said, French rail is an excellent way to get from region to region. Paris sits at the center of France's rail network.
However, rail service is often spotty in the provinces and wheels are either helpful or essential. In the summer there is a network of buses that run to many destinations in Provence, Burgundy, and other regions. If you absolutely can't rent a car, these may be your best bet.
Cycling is absolutely an option. I saw Burgundy that way on a bicycle rented from a Rick-recommended shop in Beaune. But you are subject to weather, terrain, and your own limits.
Finally, however, I wonder if this is really a good plan. London, Amsterdam, Paris, and day trip from Paris is a full two weeks, and you might lose more than you gain by shoehorning another destination into that itinerary.
Have a great trip!