Discover your interests: Think back to papers and assignments you enjoyed in your undergraduate degree for common themes. In a graduate degree, you work will concentrate on a narrow topic, so it is important that you choose a general field you find interesting. Otherwise, you can expect burnout. When looking for schools, remember that you can change your discipline. If you were a biology major in your undergraduate, you could be a sociology graduate student (there may be extra courses involved).
Choose a school and contact potential supervisors or faculty members: The school and the faculty are equally important. After graduation, the reputation of the school and its faculty reflect upon you. A well-connected school and supervisor can help create opportunities such as publications and jobs.
Start the application early and apply to multiple schools: Applications take time and require documents, such as reference letters. Starting early ensures you don't waste time and money. Don't put all your eggs in one basket, as the saying goes. Admissions vary depending on the quality of other candidates and available supervisors. You could be very qualified and still be turned down.
Survive course work: Nearly all masters programs in North America and many in Europe require graduate students to complete coursework. If you are a doctoral student, this will vary by school and location much more. Coursework is tedious, and the expectations are higher than upper-level undergraduate courses. There will be a learning curve, so be patient. If the courses offered do not match your interests, talk to the professor. There is often more flexibility at the graduate level.
Design your research proposal: The reason for graduate school is to produce a major piece of research, such as a thesis or a dissertation. Even course-based masters programs require a lengthy paper. Your research proposal will outline what you intend to study and how you will conduct the research. The research question, hypothesis and methodology are critical and will depend on your discipline. As you do your coursework, jot down interesting ideas that could make a narrow research project.
Carry out your research and write your thesis or dissertation: Conducting research is fun. After surviving coursework, you can focus on a topic interesting to you and collect original information. Writing can be a chore, and good time management is fundamental.