Try to have all your letters for application materials sent out by Mid-November. Take the Graduate Record Examination as early as possible in the Fall semester. If you are rusty in math or some other area, do one of the GRE study courses. Acceptable scores vary from school to school. Most use them as only one indicator of your probable success in grad school.
Most schools decide early in the Spring Semester about admissions.
The longer you wait, the greater your risk of not getting in where you want to, or not getting in at all. Many schools have quotas based on the number of faculty members and how many students they can handle. Top notch students sometime fail to get in schools for which they are more than qualified just because of this approach to admission. Then you wait. We recommend that you apply to at least five graduate programs.