Quick Tip: The Student Visa Process
I've gone over a few ways to stay in France long term but I'm going to give a bit more information on the student visa process as that is what I am relying on to stay in the country long(er) term.
So you can choose either the short term, if you don't want to live for more than a few months, or the long term, but if you choose the short term you will not be able to extend it once in France. You would need to go back to America to apply again.
For the short term you will need a letter of enrollment from a French school, a letter from your home university stating you're enrolled, a letter of financial guarantee (either a statement from your parents, notarized, saying they will provide you with $820 a month or a letter showing that your tuition and room and board has been paid), travel itinerary (your plane ticket), and you will need travel insurance. If you go to insuremytrip.com, click on the one specifically stating "embassy visa" as that is the one that has all of the requirements needed by the Consulate.
You will then take all of that with you to the Consulate, along with the application, copies of everything, and a picture for id. You will also need a prepaid self addressed envelope if you cannot go back to pick up your passport.
For the long stay visa you will need to go through a lengthier process. First you will need a letter of enrollment from a French school. Once you are accepted, then you need to create an account on Campus France so that they may provide you with the attestation. For my school the process to receive the letter takes three weeks, apparently Campus France takes a further three weeks to process, and then of course once the appointment is made at the Consulate that can take up to 21 days, so I guess we'll say three more weeks. All together you're looking at a little over two months. And of course it's a bit of an expensive process. Be aware of that when taking into account how much you are willing to spend altogether. The enrollment in the school can run from €90 onwards, Campus France is $100, and then the cost of applying for the visa is about $130. That doesn't include the cost of your expenses once you're there (like the airfare to or room and board).