You have a number of options for structuring a research experience within the context of your medical studies.   It can take the form of brief independent study, a summer experience, a longitudinal experience throughout medical school, or a second degree program (Master's Degree or Ph.D.). 
Options for students interested in more rigorous research training involving significantly greater commitments of time include:

Medical Student Research Fellowship: for students interested in spending up to a year enhancing their medical training through direct participation in basic science, clinical investigation, or health care delivery research.

T32 Pre-Doctoral Clinical Research Training Program: for junior trainees interested in spending one year developing skills required for a career in multidisciplinary clinical and translational research relevant to human health.  This program provides the student with an opportunity to obtain an M.A.S. in Clinical Research

P.S.T.P.: the Physician Scientist Training Program (P.S.T.P.) is a dual degree program (M.D./Ph.D.) designed to rigorously train physician scientists who anticipate an academic career in their chosen research field.