When I shadowed physicians, usually I just hung back and observed, trying to understand everything I could and then asking questions as they came up. It's going to be pretty basic stuff, at least at first - how do you approach a new patient, what's the process of meeting the patient, coming up with differential, ordering tests, managing case, etc. This will also vary depending on whether it's inpatient or outpatient. I picked more things up as I went along, started asking more specific questions, but I always followed the lead of the person I'm shadowing (i.e. shut up until they talk to me).
In general though, I think as pre-meds we don't know nearly enough to gain more than a superficial insight during shadowing (but still, I would say it's interesting and worth it).
Also, does that HSDN label mean you're in high school? If so, you are waaaaaaay ahead of the game - I hadn't even considered shadowing until well into undergrad. Don't know if I would have even got anything out of it in high school.