According to the National Society of Genetic Counselors, "they work with individuals and families with a medical history or increased personal risk for a genetic condition, or with individuals or couples at risk for having a child with a birth defect or genetic condition. They provide information and supportive counseling, coordinate testing and connect families with supportive resources such as support groups and funding agencies. Genetic counselors are also involved in teaching, research, screening programs and the coordination of support groups."
Usually one goes to genetic counseling graduate school. These are generally approximately two year programs leading to a Masters degree in genetic counseling. This site contains a list of all schools offering an accredited master's degree in genetic counseling. Entry into these programs is quite competitive. In principle one can choose from a variety of majors, such as Biology, Chemistry, Psychology, or Nursing, and enter a genetic counseling program. Whatever the major, you must be sure to complete the recommended coursework for genetic counseling programs.