Context, I am an undergrad looking to take graduate stochastic calculus next semester. Unfortunately I do not have a whole lot of experience with probability theory as a whole. I've taken a first course in graduate probability theory (up to CLT and LLN) and have a decent measure theory background. I've spoken to the instructor, he has agreed to let me take the class and has also agreed to slow down the first few classes so that we (me and a couple other undergraduate students) don't immediately get lost. He suggests that I learn as much as I can about Markov processes and martingales to get myself used to the type of theorems that I will be seeing.
I have two books, Probability Theory: A Comprehensive Course by Achim Klenke, and Probability: Theory and Examples by Rick Durrett. The former of which I have read and done exercises for up to and including chapter 15 (skipping chapters 10,11,12 on martingales).
I would be very grateful if someone could suggest what else I need to be familiar with before going into the course (I'm already planning on finishing all sections on Markov processes and martingales in Klenke)? And in particular, could someone take a quick look at the table of contents of one of the books and let me know if there's any chapters I don't really need to read.