Make sure you fit the criteria: be a U.S. citizen at the time of application; have a bachelor's degree, preferably from a U.S. institution; be a proficient speaker and writer in the relevant foreign language; be in relatively good health.
Contact your school's Fulbright Program Adviser (FPA). Your FPA will be able to get you an application, as well as give you detailed information on the application procedure.
Think precisely about what sort of program you'd like to design for yourself and about how you'd describe the project to the funding committee.
Notify professors and TAs well ahead of time for any letters of recommendation you need from them.
Send in all the materials before the deadline, which is usually in late October. Apply well before the application deadline to avoid the risk of missing it.
Photocopy your application materials before sending them in, and keep a copy for your records.
Keep in mind that the amount of money you'll receive (if you get a Fulbright) will vary according to the local living expenses of the place you want to visit.