Dropping off a resume would be a waste of paper for you/Kinko's, and a waste of time for them to read. They're busy physicians.
That in mind, they're physicians who probably like sharing with potential physicians what their work is like.
For me, it was easy because I worked as a CNA on a Telemetry unit, and lots of internal medicine physicians rounded up there every day. I just picked out a DO from the list and waited for her to round, then I asked her if she would mind if I shadowed her. She was all for it, since she already was preceptoring two third-year students.
If you don't work in a hospital setting(which you should, of course) then I would recommend picking one out of the phone book, writing a nice letter explaining who you are and that you'd love an opportunity to shadow. Mail it/leave it with the receptionist up front. If they call back, then ask if it would be okay if you shadowed them while the rounded on hospital patients(you'll get to see a lot more this way). With the mailing a letter strategy, you're not annoying their answering service with calls, you're not pestering them in person; you're letting them do it on their own terms, which might make them more amenable to your shadowing.
On shadowing day:
-Dress nicely
-Don't try to fit all the questions you have about the profession into 10 minutes. Remember that it's a privilege for you to be there; they didn't HAVE to say yes. If you don't get in the way(read, "if you actually help") they might invite you back for another (hopefully a few) session.
-Don't share your life story. If they're rounding, there's probably little time for the chit-chat. Save that for a clinic day down the road after you've been with them a few weeks.
-If you have a stethoscope, bring one. My physician was having me listen to all sorts of stuff like murmurs, crackling lungs, breath sounds, etc. Bring it, but don't wear it on display like you're wearing a medal. Keep it stuffed in your pocket and only bring it out if they invite you to listen to something (If they DO, they'll probably have you use theirs, which is probably better than yours).
-Stay out of the way, but be interested. This shouldn't be too hard.
-THANK THEM profusely for their time, and at that point ask if you could do this again SOMETIME. Exchange numbers and then wait a couple of weeks before you call again. You don't want to be THAT premed kid.
