How to make the most out of a one-on-one reading course with a faculty member? I am currently at the end my undergraduate / beginning of my graduate degree in pure mathematics.

Next term, I am going to take a reading course with a faculty member in some topic that I am interested in.

I see this as an amazing opportunity, but at the same time I worry that my habits can work against me. Until now, I've rarely stayed after class to ask questions, and I've never gone to office hours. The reason is mainly that I find it hard to ask lecturers for guidance. If I have any question (arising either from not understanding something in class, or from trying to think further about ideas thought in class), I would use the internet or the library.

To be frank, I even find the whole idea a little bit awkward. I find it hard to see how I can find enough "material" to to fill several hours a weak of one-on-one discussions, and I imagine myself having trouble coming up with relevant questions or comments. Of course, much of it has to do with general shyness. But there is also the factor of "knowing how to use guidance", which I believe is a bit more under my control.

I want to make the reading course a valuable experience, and to further my mathematical development through it. What can I do in my part to make the most out of it?
Ask the professor about his/her expectations on meeting. A short meeting once a week is probably the norm (comment of Nate Eldridge). But ask. And try to schedule a regular time for it. I think that what is expected is that you would give a short report on what you learned and that you would ask questions. If the prof is being vigilant, you might need to field a few questions also, some that might be hard/impossible to answer. The intent will be more to get you to think in a certain way than to test you. 

But don't treat it as just "reading". Find a way to do some practice around the readings. If it is a well studied subject there will be exercises available and you can report on what you have done with those. In a more "state of the art" situation you will want to think about "what if..." to get a sense about where this subject might be extended. That would be for a quite advanced set of readings, of course.