Start researching graduate programs in your junior year

Graduate school applications will be due sometime during the winter or spring of the academic year, for admittance the following fall semester. In other words, the autumn semester of your senior year is most likely too late to start looking at programs, compiling your application materials, and getting them in on time. Start early and dont miss out.
Look for programs with a good reputation, but give more weight to the faculty and the research interests of the other graduate students at prospective schools. What youre looking for in an advanced degree program is camaraderie and common ground, not an arbitrary ranking on some prestigious list.
The applications are expensivesometimes $50 or $80 dollars eachso you wont be able to apply to all programs. Try to select a range of programs to apply to: choose a few big dream schools with great facilities and prestigious faculty and lots of competition to see if you cant get in. Apply to smaller programs that youd also be happy attending. Apply to as many as you can afford to give yourself the best chance.
For some fields, a masters degree will be a more appropriate subsidiary or even terminal degree. At worst, a masters degree can be an excellent primer for the graduate school life, especially if teaching assistantships or fellowships are available.