This is an interesting question. I will take a crack at it. I assume that you are a medical student now looking for LOR for residency. I guess I would start by going to events where you know these people might be. Prime networking events would be:
specialty interest groups, research days held by the department, and other events held by the department (special visiting professor lectures, etc.). Often these events have receptions paired with them for mingling. Don't be afraid to strike up a conversation with someone and identify yourself as a medical student. You may be able to build a relationship from there.
As far as what to do. You want to find out their name and what part of the department they are in, and keep track of them in a folder/rolodex at home. If it is someone you have something in common with interestwise, don't be afraid to ask if you can contact them later for career advice/research project potential/etc. and do so when you are ready to do that. I doubt anyone would ever say no. After that, it's just a matter of keeping in touch and doing whatever it is you promised them (research projects, calling, you get the idea)
Good luck
