Is that 10 days on the ground? Even if it is, it's not much time. And it is likely to be hot, especially in the south. So if you require air-conditioning, be sure to look for or email potential lodgings about it.
Are you planning to fly open-jaw, also known as multi-city? If so, you could fly into Nice and back home from Paris or vice-versa. Airline options from where you live will have a big influence on the practicality of that.
For scenic and romantic, I think it's hard to beat Paris itself with possible side trips to Giverny or a chateau or two. This list of French castles/chateaus from Wikipedia might help you decide which ones, if any, to visit: http://en.wikipedia.org/wiki/ListofcastlesinFrance. If they are important to you, seeing castles could determine a significant part of your trip itinerary. There are lots of links to more information about the regions and individual buildings.
Nice and the Mediterranean coast in general are more beautiful than most of us can imagine, but they are likely to be be cram-packed and the hotel costs expensive. Of course, that can probably be said for most of France in high season. Be prepared for it.
France is very large and you have outlined going from the far northwest to the far southeast of the country. If you research travel time, whether by train or car, I think you will find that you cannot do all the things you outlined in the places you list in the time you have. The interactive maps on this commercial website could be useful for seeing how spread out your listed destinations actually are -- http://www.linkparis.com/france-map.htm.
Using the "directions" options with Google Maps can give you a general idea of driving times if you choose to rent a car. On our month-long trip to France in 2012, we picked up our car in Reims (after 8 nights in Paris) and returned it in Aix-en-Provence (after 8 nights there with lots of driving to explore the area). The TGV from Aix to CDG was about 3 1/2 hours with no changes.
By using the DB Bahn search option (http://reiseauskunft.bahn.de/bin/query.exe/en), I found the general information below. The schedules aren't available for August yet. These examples are from 02.06.15. Please note that if you search anything having to do with dates, the European way is day.month.year.
You can't buy your tickets from DB Bahn, but I think they do the best job of showing the routes and connections and I especially like the ability to click on schedule options to see transfer details for the options that require changing trains, intermediate stops and for how long, and (my personal favorite) the map to show exactly where you are going.
Now for the examples:
You could take a fast TGV train between Paris and Nice and get there without any train changes in 5 1/2 hours. That's probably faster than you can drive.
You could take the TGV to Annecy from Paris and get there in about 3 3/4 hours with no train changes, also faster than you could drive.
From Annecy it's about 2 3/4 hours to Chamonix by bus and regional train. In that case, driving would probably be faster, but probably more trouble.
From Annecy to Nice is about 8 hours by regional train or bus and TGV and with 2 changes along the way. You might also be able to drive it faster.
You have lots of research to do, but fortunately you have time to do it. I love doing the research. It's like taking the trip twice for me, once when I plan it and once when we do it.