I best guess is that you will not be able to shadow a nurse (RN) at a hospital because of confidentiality issues, as prescribed by the Health Insurance Portability and Accountability Act (HIPAA) unless you are part of a training program sponsored by a qualified institution.
You can volunteer at a hospital and ask to be placed in a unit where patients and nurses are (you'd probably be at the nurse's station)... that way you get to see and feel the environment, but you will not be able to go into a room when a patient is being examined or interviewed. On a rare occassion, they might use your help if you speak another language and no one else is around to translate for the patient.
Nursing homes might be a bit more flexible, but I'm not sure... you'd have to check. I don't know about doctor's offices.
If you want to start working at the entry level of nursing, you should go for the CNA license/certificate. That puts your foot in the door without spending a lot of money. It also gives you a taste of what it's like. The next level up is Licensed Practical Nurse (LPN) or Licensed Vocational Nurse (LVN)... then there is the Registered Nurse (RN). After that you can advance to a variety of positions with further education.
Hope this helps! Good luck to you!