Identify ten schools to which you might apply. Visit the career center in your school and speak with counselors who may help you with university guide books and other resources. Speak with professors in the area of your interest for their recommendations. Look at the overall ranking of the institution as well as the specific ranking of the department. For each institution, identify faculty members with whom you might find an intellectual match.
Think big and apply to at least five top-notch schools. Don't underestimate your intellectual ability and skills. Don't submit a generic application to all the schools. Modify your statement of purpose to suit the specific school in question and name the professors with whom you'd like to work and why.
Look into financial options. If you can help it, don't count on taking out big loans to put yourself through graduate school. Regardless of how well-endowed any institution is, also apply for extramural funding from the Mellon Foundation, Ford Foundation, Jacob Javitts, and National Science Foundation, among others.
Ask questions. Call up the departments of your interest and ask the following questions: What does the placement record of recent graduates look like? At what rates are they getting jobs and where are they getting placed? What is the average number of years it takes to finish the degree? Can you speak with a graduate student in the department? Is the department committed to diversity?