There's so many ways to go about it.
First, as others have mentioned, get a hold of your PCP and see if they would let you shadow them or if they know someone who is fond of pre-medical students shadowing.
Second, during my time in UG, I made a big list of local doctors with their specialties and their medical school they attended. I basically went down the list and called each office and simply asked "hey so I'm a pre-medical student here at X university and I am interested in shadowing Dr. Y. I was curious as to what your offices' policy is regarding student shadowing, if Dr. Y ever allows students to observe..."
Office secretaries are quite often kind about this. Explain to them what you study, what year you are, possible availability during the week (make this as broad as possible). I would call instead of e-mail simply because I think calling is much more direct.
Lastly, as I have mentioned on these forums before. during my 4th year of UG, while playing catch with the football outside, I jammed my pinky and it actually broke out of the skin, blood everywhere. I was obviously referred to an Orthopedic surgeon in town. When I saw him for my first appointment, he asked me what I study (super common question) and I told him I am a pre-medical student, hoping to apply in the 2016/17 cycle. He went on to probe me about my interests and if I have shadowed before. I then basically told him I would be humbled to be able to observe you here in the office. He responded to that by saying, hey the office is boring, lets get you in the OR. And that's basically how I shadowed countless of ortho surgeries. That's actually also how I realized I don't think I want to do ortho... but that's a side point.
