Start off by e-mailing the doctor's you want to shadow. Something brief to introduce yourself and explain that you're premed. Most doctors have been there and understand what you need.
Next, if they respond favorably, go to their clinic and meet them in person (or else set a time to shadow/observe them by e-mail). Once you're there go by the feel of the situation. Doctors are people and they come in all sorts of varieties. In my undergrad I shadowed a dermatologist in her private practice and she explained every case before calling the patient in, then discussed it after the patient left, and also let us help out (hand utensils, hold down kids, etc.) She was great. On the other hand, I also shadowed an ER physician at the local hospital and he just wanted me to stand to a side and pretend like I didn't exist. Point is, let the doctor dictate how things work. Same for times, if you establish a rapport with the physician maybe you can come more often than with a doctor who you don't "get" (or doesn't get you).
Take notes if you feel the need to, but don't do it for the sake of putting a show of it. Ask questions if you feel they're welcome, which they usually are. I'd recommend trying to talk to the patients too, and if you get a chance, try to get an idea of what their general history is and review it in advance. Things make more sense that way. At the dermatology clinic, if there was a patient who wasn't comfortable having me around (the doc always asked), I'd step out into the waiting room and talk to the patients about what anxieties they have. What they liked about the doctor. Why private clinic vs hospital clinics. etc.
From personal experience too: Shadow different doctors. The general clinic experience remains similar, the major difference to me is what questions they ask taking histories and what kinds of requests (EKGs/Echos for cardio, EEG/brain MRI/CT for Neuro, sleep/lung func for pulmonary, etc). Braoder exposure is better, and it's also more cool to be able to see the whole range of fields in medicine and how seemingly disconnected they are by the time they're specialized. Then there are radiologists and anesthesiologists, which feel like their own field. ENT and ophthalmology also feel like an entirely different part of medicine (to me, at least, they're not what comes to mind when I think of the typical "doctor".)
I hope going into the OR counts as shadowing! It's my favorite place to be...
As for shadowing doctors with ties to schools you're applying to, I imagine that might help, but I wouldn't know to any degree of certainty.
Good luck and have fun!