I concur--this forum is for American style backpacking, called wilderness trekking by Europeans. I have traveled a lot in France, but my last trip was 8 years ago so I can't give you current info.

I strongly recommend you read a guidebook--my preference has been either Rough Guide or Lonely Planet, depending on which has the most recent edition. You can get plenty of advice over on the Lonely Planet Forum, but you need to ask specific questions, not general ones. Otherwise they'll give you the same advice I just did--read a guidebook first.

The French railroad system is a great way to get around. However, with only three weeks, I suggest limiting your stay to only 2-3 areas. Obviously one of those will be Paris and vicinity, where you can spend well over a week.

Watch out for 3- and 4-day holiday weekends occurring April through June that, at least in the US, we don't even think about--Easter, Ascension, Pentecost, V-E Day--accommodations will be full so you'll need to reserve in advance. Most schools and universities have several weeks off around Easter, adding to the crowds.

I several times had the experience of going into a hostel dorm room, introducing myself in French and discovering that everyone in the room was American, British or Australian! The one place I went where there were only French people present was a hostel in Corsica (most of the guests were from Paris and vicinity). It's amazing how much your French improves when you have to speak it all the time, even for only a week! Unfortunately, in most places, unless your French is good enough for you to pass as a native, as soon as you open your mouth most people switch to English. If you can possibly arrange a homestay, that's the best way to improve your French.

Bon voyage!
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May your trails be crooked, winding, lonesome, dangerous, leading to the most amazing view--E. Abbey