How do I design a project? Try thinking about your project as a series of questions to be explored. Are there questions in your major or minor field of study that you would like to pursue abroad? Is there some kind of creative work or academic topic (Taiko drumming in Japan, Polish Dance Theatre, community bartering networks, plant-insect relationships in extreme climates) that would be best conducted or studied abroad? Then decide what specific questions about that topic interest you the most. Setting out the questions may help you to see exactly where you need to go, as well as providing you with a starting point for the necessary explanation of your methods and goals.
How do I find an institutional affiliation? Decide on the focus of your graduate studies or year abroad and find the program that's ranked highest in your field; don't merely select an institution based on overall reputation or prestige. Be able to justify your choice. For example, you will need a more compelling argument than "I want to study in England because it's Shakespeare's birthplace." Many of the world's greatest Shakespearean scholars teach in American universities. Send off for applications and course brochures (if awarded a Fulbright you are in most cases responsible for obtaining admission to the university yourself); if appropriate (especially for scientific research,) initiate contact with faculty at the institution. Don't be afraid to send an email to a scholar you don't know. If you provide them with a brief explanation of who you are, indicate that you are applying for a Fulbright grant, and a short summary of your project, they will often be willing to discuss the project and an institutional affiliation with you. Providing evidence of "affiliation"--a letter, fax, or email from faculty expressing willingness to supervise your research, documentation that you have begun the application process--will help to persuade the committee that you have seriously considered your project's feasibility. If the project you are planning would not be best served by affiliating with a university (for example, if you are working on a public health education project and want to work with local NGOs), do some research to make sure that you know who these organizations are and be ready to talk to them about how you can cooperate with them as you do your work. Fulbright will not want to fund you to work for an organization, so be clear both with the organizations and in your application about how you will work with these groups rather than for them.
How do I pick recommenders? As with any program, you want to select recommenders who know you best, and can comment on you as a person both inside the classroom and out. You will want at least one academic letter, from a professor in your major field. Your other two (and yes, the Fulbright program is pretty firm about only including three letters of recommendation) should be from people who can best testify to your abilities to carry out the project or work you propose. These can be teachers, supervisors at work, or supervisors from volunteer projects or extracurricular activities. Think in terms of personal skill sets as well as academic abilities: can your recommenders describe you as the kind of person who would make a good candidate for an exchange program--friendly, adaptable, a resourceful problem solver, a person who meets challenges well? For more help with recommendations see the advice on asking for recommendations elsewhere on this site.