I have been living in France for a number of years, and it is great that you are coming to visit. I lived in Paris for a few years and now I am in Normandy. In Paris, there are tons of things to do that are not touristy. If you are into the local bar scene, go to Oberkampf metro station, there are lots of bars that pretty much only locals, kind of artsy types, go to. There is the St. Martin Canal, which during to summer is a popular hang out spot for young people. (It is in the tenth arrondissement, I believe). Also, the bars and restaurants around Bastille (rue de la Roquette) are quite popular. You can also walk around Les Halles (2nd Arrondissement), especially behind the huge church (St. Ouen), where there are a couple of little streets with restaurants and shops. Also, check out the market at Moufftard (6th).
In Normandy, one of the best places to go is Honfleur. Take the train to Le Havre, and just across the bay, there is a city called Honfleur. (There is most likely a bus to go there from Le Havre). Not many americans go there, but it is very beautiful in the summer time. Also in Normandy, you could go to Giverny. It was the home of Claude Monet, and there are beautiful gardens to walk around in. Only problem, it is very touristy.
There are a lot of hostels in Paris, and at least one or two in other major French cities. In Normandy, that means Rouen or Le Havre. Or Caen, possibly, but Caen is boring.
As for the trains, you shouldn't have any problems. Most French people travel by train, and France has one of the best train systems in the world. You can pretty much go anywhere in the country. French trains rarely leave late, unless there is a strike, so be on time and don't forget to time stamp your ticket (the machines are bright yellow) in the machine right before you get on the train. This validates the ticket for the trip. Also, if you get your ticket and don't use it, or miss the train, you can use the same ticket for a later train no problem, just stamp it in the machine before you use it. The ticket is valid for a certain time (I think an hour) after it is stamped.
Hope this helps, and enjoy your trip!