I am now attending college as part of the Running Start program and am planning to become a pediatrician. First, to apply to medical school you will need to earn a four-year baccalaureate degree. From your message I sense that you currently are attending a two-year college because you ask about earning an associate's degree. You mention transferring to a four-year school, so your thoughts are already on target. Following completion of a four-year baccalaureate degree, you will need to apply to medical school.
Medical school is another four years of study.
Following graduation from medical school, new doctors are required to participate in a residency to further their skills. Residency positions can last anywhere from three to ten years, depending upon one's specialization. Another issue you raise is the academic requirements for medical school. I can give you a sense of the topic areas you will need to take courses in, but I suggest that you arrange an appointment to speak with a pre-professional advisor at your current college. You may also need to take a formal biochemistry course, however I again suggest that you speak with a pre-professional advisor who can outline for you the current requirements and recommended courses for particular medical schools.