This advice is especially suited for students searching for a Master's or PhD program in ecology, evolutionary biology, conservation biology, or a related field in biology, including students interested in joining my lab. Much of the advice also applies to most other fields in the natural sciences. I suggest approaching the process in 5 steps, as described below.
Step 1: Plan to take the GRE. More and more graduate programs in biology are no longer requiring the GRE, so it may be possible to avoid taking it, but you may be limiting your options if you don't take it, as many programs still require it. I suggest planning to study for it, and completing it, more than one year before you want to begin graduate school (e.g., during the summer before your senior year in college, if you want to begin a graduate program in the fall after you graduate from college). Most importantly, keep in mind that you will need to take the GRE at least 3 weeks in advance of your first application deadline, and many graduate programs have application deadlines in November or December, for admission the following fall. Taking the GRE in the summer will allow you plenty of time to prepare and get it completed before the busy fall term starts.
Step 2: Finding potential advisors and programs. Recognize that the most important aspect of your experience in graduate school will be the lab you join, and the professor who is your advisor. A secondary consideration is the overall reputation of the university and department, especially in your field. Start by finding faculty who are doing research that is exciting to you, at a diversity of universities. How do you find them? Some possibilities: a) consider the authors of scientific papers or books you find exciting, b) ask your closest academic advisors for advice (they usually can recommend some people), and c) peruse the websites of relevant departments at good universities in parts of the world where you might want to live.
Contact those professors, expressing your interest to work with them in a concrete, specific way (ideally, referencing particular aspects of their research that you have been inspired by). Tell them about your experience and interests, and ask if they are accepting applications from prospective graduate students. Regardless, at this stage, you should endeavor to find at least several good options, so that you can eventually narrow your choices down to the very best option for you. I suggest initially contacting around 10-12 professors. Based on their responses, you can begin to narrow it down to 3-4 good possibilities. Some will not answer, some will give negative responses, and some will be positive. I recommend beginning this process at least one year before you want to begin graduate school. You don't want to be writing to potential graduate advisors immediately before the application deadlines at their schools, which can sometimes be in November or December (for admission the following fall), because they may already have established strong ties with other potential graduate students by then. It is better to get yourself onto their radar a bit earlier (**although if you are a good applicant, they will likely still be interested, even if you contact them later than some other applicants, so don't ever think it's too late, if you find someone who might be a good fit as an advisor).
Do not over-react to any particular positive response, and resist the temptation to lock into one good option. Putting 'all of your eggs in one basket' is too risky. At this stage, it's important to keep in mind that not only are you trying to find the best graduate program for you, but professors are also likely fielding inquiries from multiple prospective students. They likely cannot, and will not, ultimately accept all the students who want to work with them. So, you may be competing against other students who want to join that same lab group. Even if a professor agrees to support your application to their program (which can sometimes be a prerequisite for your application even being considered), and encourages you to apply, that does not guarantee that you will be accepted to the program. While you are getting to know multiple potential advisors, those advisors are getting to know multiple prospective students. In addition, your file will be reviewed by an admissions committee to determine your qualifications and rank you against all applicants applying to that same program, including those who want to work with other professors in that same department. If your application does not meet the minimum criteria for entrance into the program, or rank highly enough among all acceptable applicants, your chosen advisor may not have the choice to accept you. Hopefully your top choice of programs will accept you, but you cannot count on it. This is why it is important to have multiple options. Indeed, it may be smart to include some choices that are more of a "sure thing." You can always ask professors about your chances of admission, and whether they are also considering other students. They may ask you also if you are considering other programs. Open, honest communication is always a good idea, especially as application deadlines approach, or deadlines for accepting offers of positions.
Step 3: Narrowing it down. With professors who give you positive initial responses, you should begin a professional dialogue with them by e-mail. In this correspondence, you should be prompt, courteous, and always professional. Remember that these e-mails are your first chance to make an impression on these people. The more you can exchange ideas and opinions about science, the better, but you will also want to ask practical questions about their department, graduate program, funding possibilities, and what are the professor's expectations for students in their lab group. Note that some programs accept students into individual labs, some ask you to do a rotation among several labs when you first arrive (before choosing one), and some (like the University of Mississippi Department of Biology) have both options. If there is a rotation system only, make sure there are several good options for potential advisors in that program, and communicate with them in advance to make sure they would be willing to have you rotate in their lab. A phone or Skype call is recommended at some point, to give you both a chance to ask and answer a bunch of questions in short order, and get to know each other a bit better. Make sure to prepare for such conversations by preparing questions to ask. Get advice from trusted professors and/or graduate students, regarding what questions to ask. Make sure to ask about funding that the program might be able to provide, in the form of teaching assistantships or fellowships, and ask the professor whether they might be able to provide research assistantship funding from a grant. Ask them to provide names and e-mail addresses from former members of their lab, so that you can write to them and ask about their experience.
US citizens should also strongly consider applying for a National Science Foundation Graduate Research Fellowship (https://www.nsfgrfp.org/). Applicants in other countries should seek other fellowship application opportunities (such as Fulbright). Make sure to discuss these fellowship applications with your prospective advisors, as some may have ideas and/or be willing to help with your application.
Visiting your top choices is highly encouraged, if possible. This can be done during this stage, if you can afford to pay for your own travel, and you can also ask if they would be willing to (financially) help you visit. Some professors or programs will pay for you to visit before your application is submitted; if this happens, it is a good indication that they are very interested in accepting you. Other programs will host a visit day (or weekend) for their top applicants after the application deadline; this is currently what happens for PhD applicants to the Department of Biology here at the University of Mississippi. Being invited on such a visit is a very good sign that you will likely be offered a position, although it is not a guarantee. International applicants, even those accepted to the program, are not always invited to these visits, because it would often be much more expensive. Of course, if you can afford to pay for your own visit, that is often an option.
Stage 4: The application process. Read instructions very carefully for each program to which you intend to apply, and do this well in advance of their deadlines, so that you will be sure to have everything in order when the deadline comes. You should try to submit all of your materials at least one month before the application deadline. This will give your potential advisor(s) time to review your application materials well before the graduate admissions committee. If your application looks good, this can give you an advantage. If there are issues in your application, this can give your potential advisor an opportunity to tell you about them and give you feedback. Most applications will require some kind of 'statement of purpose.' This statement should be as research-oriented and professional as possible; strong, concrete research ideas (to the extent that you have them) should ideally be included, as well as concrete summaries of your different research experiences (and any products from them such as posters or talks at a conference, manuscripts being drafted, or best of all, publications) and how they have led you to be interested in working with this particular professor. As such, you should get advice and feedback on this statement from one or more trusted advisors, and sometimes the professor you are applying to work with may be willing to give you advice and feedback on it.
Step 5: Making your final choice. Hopefully at this stage, you have multiple good options, having received official, written offers of admission from multiple schools. Don't get too excited or sucked into any one possibility before you know what all of your options will be. Try to wait until you have all the information you need, and then make the best choice you can. You should consider your rapport and level of trust with the advisor, his/her history of success in training and graduating students, feedback from former members of their lab who had good experiences, the fit of your research interests with the opportunities presented in that lab and program, funding opportunities (including amount and especially availability & reliability of assistantships and fellowships), the overall quality of the program and university, the other faculty and students in the department, and the quality of life in their city. Talk with trusted advisors about your choices. Visit your top choices at this point, if you haven't already, and if you can afford it (or if they will pay for it). Make your choice. Always be prompt and professional in all of your communications.