Memorizing proofs doesn’t really do much for you, at least in the long run; instead, you should try to see what makes them tick. First, what is the structure of the argument? What are the main steps, and what are merely details of carrying out those steps? Many proofs at this stage of your studies have just a single main idea, and everything else is details. Secondly, what kinds of details appear over and over? What basic technical tricks keep reappearing? Those are tools that you want to master for your own use.
Here are some books which I think can be classified as 'reference books.' Some are less rigorous than the material you'll learn in advanced pure math courses, but will always be helpful to you. I say this as an opinion, as they have been helpful to me throughout my university career.
I didn't cover all of the areas you listed but hopefully this helps.
Abstract Algebra - Dummit and Foote, Abstract Algebra. This book is humungous and contains pretty much everything you'll need in undergraduate algebra.
Calculus - For single variable stuff I say Spivak's Calculus. For multivariable, I still look back at James Stewart's Multivariable Calculus. Although this isn't a pure math book, it contains all of the main theorems, and more, you'll learn in 2nd year. It also contains tons of examples. When you learn differential geometry you'll go back and look at these theorems in a different light. A more rigorous multivariable calculus book that is worth storing on your shelf is Spivak's Calculus on Manifolds.
Real Analysis - I think J.
Marsden
and
M.
Hoffman,
Elementary
Classical
Analysis is a great reference.
Linear Algebra - Friedberg, Insel, Spence, Linear Algebra. I still use all the time.
Complex Analysis - L.
Alhlfors,
Complex
Analysis contains all the fundamentals.
Topology - Munkres, Topology. I think everyone who has studied topology knows this book. A definite reference book to have.
Best Answer
I think the best thing for you to do is to talk to your academic advisor if you haven't done so already. The reason I say that is my guess that an actual professor will have more information on:
That is assuming you have an advisor who is relatively easily accessible and who will be willing to spend time on a brief conversation with you. This is the best case scenario, as in addition to aforementioned advantages related to the access of information, talking to the advisor gives you an option for real-time dialogue with the person advising.
If, however, talking to academic advisor is not a viable option for you, I would recommend taking the courses you listed in the following order:
I assume you already have solid background in (multivariable) calculus and linear algebra. If not, you should take these to ASAP, and definitely before you take anything from your list.
Also, I want to point out again that whatever I wrote should be taken with a grain of salt, as it is intended to be a "rule of thumb" list. In order to make proper decision on which classes to take when, you should take into account a lot of individual-specific factors, included but not limited to