Studying Graduate Level Mathematics Outside Lecture Hours

You need to be able to do all of it really, both the problems and understanding the proofs from classes. You'll just need to prioritize as you go.

I always went over lecture notes after the class to make sure I understood the proofs and formulations, however I didn't re-write it all as I was doing this, I would just add some extra explanation at parts where I felt I needed it. What I would do was to make sure I understood the lecture notes and then do the problems, then once the whole course has finished I would go right back through and re-write the proofs and notes, but in a condensed form if I could.

Just to make sure I am being absolutely clear you definitely DO need to make sure you understand everything that happens in lectures, often with solving problems in maths we need a thorough understanding of the subject. I have known plenty of people who just try and apply theorems without understanding the content of them and it never ends well.