Meeting With an Alumnus: How to convert a meeting into an opportunity?

Second year student at a target here. I have arranged some meetings over coffee with some alumni from my small college prep school over the course of the next few weeks, and most of them are in private equity. Obviously I am trying to network and build contacts for possible internships in the future, but I don't want to look so keen and eager that I appear superficial. I want to propose our meetings as merely a "fellow [school] alumni interested in your field and wanting to meet you and ask a few questions" rather than simply barrage them with questions intended to demonstrate my knowledge of finance and impress them with the obvious goal of "landing an internship". Do you have any ideas for talking points that may indicate my interest without pushing too hard? More importantly, any suggestions for introducing the possibility of a future internship without sounding like I am asking for too much or "using" them, or should I even introduce this idea at all?