Apply for a Schengen visa at the French consulate nearest to you in the United States. This is a short-stay visa for the 24 European countries in the Schengen region, which includes Austria, Belgium, Denmark, Finland, France, Germany, Greece, and Italy. It is absolutely imperative to have a Schengen visa in your passport before arriving in France; the Prefecture de Police -- which handles all visa requests -- will not even let you make a request for a titre de sejour without one.
Buy your ticket to France once your visa has been approved. You will have to establish residency in France to be able to apply for citizenship, and that means moving there. Do not purchase your ticket before getting your Schengen visa in case you run into any problems at the French consulate.
Rent an apartment and set up utilities. You will need a valid address and electricity bill in your name to obtain a long-stay visa with working papers, which is necessarily to apply for citizenship. You will also need proof of integration into French culture to become a citizen, so get everything in your name.
Prepare your documents, including a birth certificate that was printed within three months of your application. The French bureaucratic system is very complicated and requires many documents, so read the list that applies to your situation carefully before going to the Prefecture. Make photocopies of all of your documents and get ID photos taken beforehand in a government-approved Photomaton. You can find Photomaton machines in various metro stations.
Make a Premier Demande d'Un Titre De Sejour at the Prefecture de Police in Clichy. This is a first request for a visa, similar to the American green card. To obtain a titre de sejour with the right to work, you will need to be sponsored by an employer or have immediate family permanently residing in France, such as your parents or spouse.
Maintain a residence in the country of France for at least five years, four if you are married to a French citizen. You cannot have an interruption in your residency if you want to obtain French citizenship. You will need to renew your titre de sejour each year during this period. It will help your citizenship application if you open a bank account in your name and register for social security.
Drop off your "demande de Acquisition de la nationalité française" at the Prefecture de Police for your neighborhood. The required paperwork is different depending on your situation, which may be naturalization or marriage, but you will need photocopies of all of your documents.