I'll be doing my first research project with a professor and although the details are a bit unclear, he gave me the topic at hand and the problem we'll be tackling: the Sitnikov problem.

To quote him,

He also suggested that I should learn about the following topics:

I'm currently working from Introduction to Hamiltonian Dynamical Systems and the N-Body Problem by Hall and Meyer. I read like the first 50 pages, then I just started picking the book apart for the stuff in the latter quotes. However, some of it is a bit unclear and some of the topics are not in the book (these are boldened), thus leaving me searching for more sources.

I was wondering if anyone knew of any good books/pdf/article/etc. on these topics, or perhaps on the Sitnikov problem. I could use the help.

Vladamir Arnold's Mathematical Aspects of Classical and Celestial Mechanics (NOT to be confused with Mathematical Methods of Classical Mechanics) is, by far, the most authoritative text on mechanical systems. I think he covers all of the stuff you bolded.