While your enthusiasm is admirable and your idea sounds reasonable, it may be more difficult than you think to arrange the type of experiences you may be envisioning. There are a variety of legal and administrative issues that are making it increasingly difficult for people to "drop in" to a hospital (or other health care agency) and observe in an informal matter. Below, I will try to point out a few -- not to be discourage you too much, but to provide you with the information you may need to be successful at arranging something.
1. HIPAA -- All health care agencies are subject to this law (that went into effect 4/14/03) that prohibits them from sharing health information with anyone who doens't have a legitmate need for it for the care of the patient. Students in teaching institutions who are participating in recognized teaching/learning experiences are considered a reasonable exception. However, a hospital might well be skittish about arranging something independently with you. They may understandably want you to have the "legitimacy" of coming to their facility through a course or program through your school.
2. Malpractice insurance
. Who will be legally and financially responsible for any malpractice suits that might arise during your experience? If you are an employee of the hospital, they will cover you. If you are coming as a student from your school -- the school will need to sign papers saying that they will cover you. Few good hospitals will allow people to just "come on their own" without some kind of legal/financial agreement in place.
Based on the 2 points above, you might want to talk with your academic advisor at the school to discuss whether the school will sponsor you for anything special you might want to arrange.
Also, if you have a special interest in a particular area that you won't get much time in as a student, talk with your academic advisor and/or other faculty members about it. They might be able to get you in somewhere for a little extra experience there as part of an existing class.
What about volunteering? You might be able to volunteer with a minimum committment of hours. (I imagine that the part-time job option might be a problem because of your school committment.) As a volunteer, you would be covered by the hospital.
Just a few thoughts from someone who works as a liaison between schools and my hospital for a living.