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First you should narrow down what position you want to shadow. Do you want to be a nurse, a doctor, a Physician's Assistant, a Physical therapist, a Nurse Practitioner, etc... Then decide what setting you'd like to shadow in such as a Doctor's office , an emergency room, a surgery suite, etc.. Also some hospitals will have programs for pre-med or pre-nursing students to help them gain volunteer or shadowing experience. You will need to call the office and ask if they will allow you to have a shadowing or internship experience. It's as simple as a phone call! Don't get discouraged by being turned down by one place and try again. Be polite, tell them your interests and be available to shadow at least half a day. For internship information I would contact your advisor or career advisor at your college, they will have more information on opportunities.
The advice the Elizabeth below me gave is spot on. I just wanted to add that if you find you aren't having much luck in contacting places yourself, you can get in touch with the admissions office of a school near you that trains which ever field you are looking into. They keep list of alumni and usually have an idea of who is willing to take on shadows. You can also call volunteer services at a bigger hospital (especially if it's a teaching hospital), they usually have more time to talk to people than receptionists and can usually point you in the right direction.
Best of luck!