I've shadowed ER doctors quite extensively (hundreds of hours), a neurologist for a decent amount of time, and a pulmonologist for a little bit. Of the three experiences, shadowing ER doctors has definitely been my favorite. I've seen a wide range of different patients, symptoms, and procedures. I recall seeing a full arrest on my first day, I remember seeing a poor kid with a rare neurological disorder (I think it was called Ohtahara Syndrome?), I remember seeing a lumbar puncture, lots of ultrasounds and EKGs, lots of interesting psych paitients, and just the other day I even helped with a paracentesis and saw a tooth avulsion. Neurology and Pulmonology were certainly not as interesting.
I'd like to shadow an immunologist, endocrinologist, pathologist, a cardiologist, and an infectious disease doctor if I can get the chance.