What might you do in a typical first session with a student?

I think the most important part of tutoring is mutual understanding. It's always frustrating to have a tutor that goes too quickly through a concept you don't understand. Likewise, it's not very beneficial for a tutor to waste time on a subject you already have to a good grasp on. Getting to know the student and allowing a student to be able to speak up about what he/she does and does not know would be a goal of my first session.

How can you help a student become an independent learner?

A problem I see with many students is that they know that they are stuck, but they have trouble pinpointing where exactly to start. I think the Internet has fantastic resources to supplement learning for students in almost any subject. In order to become an effective independent learner, students must be able to draw connections between multiple concepts that build upon each other and see how one example problem can be applied to other exercises.