I'll be taking the next year off, and I intend to cover some mathematics material out of class under the supervision of an instructor. For instance, I have decided to cover both complex analysis and algebra over the summer. To that end, I am looking collect a good bunch of references that meet the following criteria:
The first book should be a textbook that will introduce me to the basics; the textbook should include computations, and introduce the subject comprehensively, at the level of an undergraduate course. Also, I should also be able to use such books to prepare for the GRE.
The second book should be a good supplement to the first book. By keeping the more advanced book on the side, I should be able to understand the theory deeply, and study the subject in more detail.
It'd be great if someone could recommend good and approachable graduate level textbooks as well. The textbook should be approach, and a good companion to other resources for the self-study program.
I, for instance, have preliminarily decided to keep the following textbooks:
Complex Variables and Applications by Churchill; and
Complex Analysis Gamelin.
Functions of One Complex Variable, Volume I by Conway.
Abstract Algebra by Dummit and Foote; and
Basic Algebra by Anthony Knapp.
Please feel free to suggest a good bunch of textbooks which one can use with each other to study a subject more thoroughly than can be taught in a course. Ideally, as part of such a study plan, I intend to not only cover the basics also to cover the theory in greater depth. So, in addition to covering material from books in the undergraduate book(s), I am hoping for some suggestions for graduate level textbook(s) that can be used on the side with the undergraduate level books as part of an extensive self study plan.