Here you'll find a rough timeline indicating when I went through the different steps in the application process. I spread everything out over about six months to make sure that I wasn't rushed, and I think this is the way to go -- it doesn't eliminate all the stress from this nerve-wracking time, but it helps keep things manageable.
I took two years off between graduating from college and starting graduate school. During "application season" I was working full-time as a research assistant in a lab, and thus I had a fair amount of free time during evenings and weekends to work on applications. If you are applying while still attending college, be advised that you may want to give yourself even more time, since you will have a lot of studying, homework, and extracurricular activities to contend with as well.
June - Begin making a rough list of schools where you might want to apply. Sign up to take the general GRE over the summer -- this will give you time to study and retake the exam before application deadlines if you don't do as well as you'd like. If several of your potential schools require or strongly recommend a subject GRE, sign up for that as well. (I didn't take one.) Purchase relevant study guides. ETS promises to send you a GRE prep CD-ROM when you sign up for the test, but this often takes months to arrive -- don't rely on it. I used a Princeton Review book, but you can choose from many study resources in print, software, or website form. You can even sign up for a GRE prep class through Kaplan or a similar company, but these can be very expensive.
July - Devote a block of time each day for graduate school preparation. This could be studying for the GRE, researching schools, reading papers from journals in your field, going to networking events and conferences, or anything else that might help you to prepare. I spent most of this time on GRE prep and reviewing university websites. When examining potential schools, I looked at their curricula, faculty within the department, rankings in my field, signs of administrative competence, and location. More information on how to choose a school (before you apply, and in the happy case of being accepted to and choosing between multiple programs) can be found on my Choosing a Graduate Program page.
August - Take the GRE (earlier or later is also fine). Continue reviewing programs. Narrow your list of potential schools down to a final dozen or so options. If possible, go beyond the websites here -- talk to faculty mentors, friends who may already be in graduate school, and other people who can give you some personal opinions about the places that interest you. Spend this time researching the top contenders more thoroughly to figure out which will make the application cut.
September - Online applications usually become available around Labor Day. Check out the applications for the schools where you plan to apply. Create application accounts, whether through the university proper or a third-party application service like ApplyYourself (this varies from school to school). Make sure to keep track of your usernames, applicant ID numbers, and passwords! Familiarize yourself with the online applications and determine whether any parts will need to be completed on paper. Take note of how many letters of recommendation, essays, and supplemental forms need to be filed for each school. Keep track of this information in an organized notebook or spreadsheet where you can check things off as they are completed. Concerned about all those application fees adding up? Many schools offer application fee waivers -- check the admissions websites for more information about how to apply for a waiver if you need one.
October - At this point you should have finalized your list of schools. Request the appropriate number of official transcripts from the registrar offices of every university you have attended. Begin thinking about your application essays and work on first drafts. Contact professors, supervisors, and other professional or academic colleagues about writing letters of recommendation. A concise, polite email to each person should be fine to start -- remind them of who you are (if it's been a while), tell them that you're applying to graduate school, and ask whether they'd be willing to write you a recommendation. Offer to meet with them or set up a phone call to discuss your plans in more detail, if they wish. If they agree to recommend you, prepare some materials to help them with their letters: an unofficial copy of your transcript(s), copies of any significant work you completed under the recommender's guidance, a draft of your statement of purpose (if you have one), and the list of schools where you are applying (with their deadlines). Get these materials -- as well as any other forms that your schools may require -- to your recommenders as soon as possible.
November - Revise your statement of purpose and any supplemental essays. Have a trusted friend, professor, or university writing center volunteer go over the essays with you and suggest improvements. Take them to heart. Be cognizant of the different word limits for these essays -- some schools want 500 words; some give you 1000 or more. Some want personal and professional information in one essay; some prefer for you to create separate professional and personal statements. Prepare great essays of different lengths if needed, but be sure that information included in the longer version is adding depth as well as length. Stay in touch with your recommenders, but don't nag them too much. I asked my recommenders to submit their letters before Thanksgiving, because my first application deadline was on December 1. If yours are later, you can wait longer, but try not to let them go until the last minute. Begin filling out application forms and assembling packages of paper materials for each school, if needed.
December - Finalize your applications. I tried to have each one ready a week before the actual deadline, so that if any problems arose I had time to fix them via postal mail if necessary. Don't put things off until the last minute! The forms may seem straightforward, but can be quite time-consuming as you answer the same questions over and over again, calculate your GPA in different ways, and reformat your essays and attachments to suit each school's requirements. Many schools have a deadline of December 15. While some deadlines might be later, I suggest trying to complete all of your applications by this time. A few days after submitting each application, I emailed the department administrator for each program and asked if they had received my materials. If there are any problems, correct them as soon as possible. It's better to find out and take care of these things before the department breaks for the holidays! Hopefully you'll have everything turned in by your winter break, and will be able to relax and enjoy the holidays without deadlines hanging over your head.