How do I study Stein & Shakarchi's Complex Analysis

Generally, one takes a "complex variables" course before taking "complex analysis", just as one takes calculus before taking real analysis. So it kind of depends on what your goal is. If you want to learn how to deal with complex functions, like taking derivatives, path integrals, infinite series etc. I would buy a book with "complex variables" or "theory of functions of a complex variable" in the title. A complex analysis book is going to be driven more towards theorem proving. Generally the proofs you get in a calculus class (like calc I, calc II, calc III) are not really proofs at all but more like proof-sketches. In general it takes an entire semester of real analysis (sometimes called advanced calculus) to prove all the results in a semester of calculus. For instance, in real analysis I you generally prove all the results from Cal I. I can't speak to your class in particular but there just really isn't enough time in a semester to do all that. Generally they teach you how to use calculus first, as that's all a lot of professions need, and then they leave the rigorous development of calculus to later courses. It's similar with complex functions. You've bought the real analysis I analog of complex functions and not the Cal I analog of complex functions. The author is going to assume you are very experienced in mathematical thinking/proofs and is going to make leaps in certain areas based on that assumption. If your goal is theorem proving, I would buy a book on real analysis first ( I probably wouldn't go with rudin), as certain results from it are going to be pre-requisites in a sense, and possibly a book on theorem proving. "Mathematical thinking: Logic and Proofs) is a good one. In general Math Majors will take a "proofs course" on how to write proofs and proof methods/strategies prior to taking theorem driven courses like analysis and abstract algebra.

I should add that as a math major I maintained a perfect gpa in my classes with virtually no attendance. I just read the book. So it is certainly doable. That is, you don't need to actually take the class in order to learn the material. I wouldn't get discouraged in that sense. The one nice thing about learning it at a university is the sequence/order in which you should study certain material is laid out for you. It's going to be extremely difficult to follow a complex analysis text without having learned real analysis first.