To learn linear algebra with a focus on proofs and not calculations, I recommend Linear Algebra Done Right by Sheldon Axler. The approach this book takes is to avoid using determinants as much as possible, which I agree with. (Note: I used the second edition of this book, the third edition looks quite different.)
To be sure, it is important to know how to calculate determinants of matrices, but this is usually done before proof-based linear algebra anyway in either precalculus or a lower division linear algebra class. The theory behind determinants, however, is more advanced, involving exterior algebra, and should be learned after one has some facility proving things about vector spaces and linear maps.
Algebra (first course)
A Book of Abstract Algebra, Pinter
Galois Theory
The standard, rigorous treatment is Lang's Algebra Ch. 5-6. Galois Theory Through Exercises by Brzezinski is a very nice problem book. Neither of these is very friendly to a beginner though. Maybe the best introduction is Cox's Galois Theory
Analysis (first course)
A good book to first learn the subject is Elementary Analysis by Ross.
Principles of Mathematical Analysis, Rudin; reviewed here. Not lightly recommended. Maybe not seriously recommended at all.
Point-Set Topology
Topology, Munkres, or Introduction to Topological Manifolds, Lee
Manifolds
Introduction to Smooth Manifolds, Lee
Commutative Algebra
The two most well-known books on commutative algebra are Atiyah-Macdonald's Introduction to Commutative Algebra and Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry. I love both of these books, though they have opposite flavors. Atiyah-Macdonald is sparse (not much more than a hundred pages!), contains little explanation, has very concise, elegant proofs, and lots of good exercises. Eisenbud's book is gigantic (almost 800 pages), much more uneven, and is filled with copious explanation, long digressions, historical anecdotes, and extensive appendices. It is an amazing book, endlessly fascinating.
For most people, commutative algebra is not an end in itself so much as a stepping stone on the way to algebraic geometry. Eisenbud is written explicitly to be a prerequisite to Hartshorne, and it proves all the results Hartshorne assumes, but it also covers much more material, well beyond what one needs to start algebraic geometry. As tiny as it is, Atiyah-Macdonald does cover most of the important topics you need for algebraic geometry, except for the theory of Kahler differentials and homological algebra.
Algebraic Geometry
In a modern mathematical education, students of algebraic geometry first learn classical algebraic geometry, the theory of algebraic varieties, and afterward learn the modern apparatus of the theory of schemes (more complicated objects that generalize varieties), developed by the French school in the 1960s and first given a systematic exposition in Alexander Grothendieck's EGA.
Hartshorne's Algebraic Geometry was written to be a more digestible version of EGA. Part of the way Hartshorne accomplishes this is by relegating a huge amount of material to the exercises, in which a substantial portion of the theory is developed. There are many flaws in Hartshorne, in terms of notation and otherwise. Chapter 1 covers algebraic varieties, Chapters 2 and 3 cover schemes and their cohomology, respectively, and Chapters 4 and 5 apply the theory to curves and surfaces, respectively.
These days a lot of people use Vakil's notes, which I like. It has an interesting style, which I would describe as the opposite of sparse. It says it all, tells you everything, even if you maybe didn't want to know.
Arithmetic Geometry
Rational Points on Varieties, Poonen
Algebraic Topology
Some people use Hatcher's book, but I'm going to recommend alternatives to Hatcher here. The friendliest introduction is the second half of Munkres's Topology, but this only covers the fundamental group and covering spaces. Rotman's book is a decent introduction which properly emphasizes algebraic aspects and functoriality of the constructions; however, it is written sort of out of order.
Alternatively you can try May's Concise Course. This book can be a bit of a thrill ride: you make of this book what you choose to. The long proofs are probably best skipped, but the main takeaway is the approach and perspective. Your eyes will be opened and you will see homotopy theory everywhere. Moves rapidly in the direction of modern algebraic topology, toward spectra and extraordinary cohomology theories.
After a first course in algebraic topology which introduces the fundamental group, homology and cohomology, a great next step is reading Milnor-Stasheff's Characteristic Classes. This has cool applications to embeddings for example. It's a nice book with cute sketches.
Derived categories
Fourier-Mukai Transforms in Algebraic Geometry, Huybrechts. Also, do these exercises (from my advisor).
Toric varieties
Toric varieties is an amazing subject: roughly speaking, it is the study of varieties defined by monomial equations (e.g. equations like xy = z^2, with monomials on both sides). What is remarkable is that these varieties admit a combinatorial description which allows one to do calculations in algebraic geometry by manipulating polytopes, rays, cones and fans.
The crash course on toric varieties is Fulton's book and the big book is the tome by Cox, Little and Schenck (>800 pages).
Logic & Model Theory
Model Theory, Marker
Set Theory
Set Theory by Jech is a good graduate-level introduction.
Physics
Quantum Field Theory: A Tourist Guide For Mathematicians, Folland
Quotes about books and reading
To me, there is nothing harder to read than a math book (papers included). To read through a math book of hundreds of pages, from beginning to end, is a Herculean task. When I open a math book, there are first axioms and definitions; then there are theorems and proofs. I know that mathematics is a thing which becomes incredibly easy and clear once you understand it, so I try to read only the theorems and somehow understand. I try to think of proofs on my own. Most of the time, I don't get it even after I think about it. Having no other choice, I try reading the proof in the book. But even after reading it once or twice, I still don't feel like I understand it. So I try copying the proof into my notebook. Then I notice a part of the proof I dislike. I try to think if there must be another proof. It's great if I find one right away, but otherwise it takes a long time until I give up. And if I go about in this way, after a month finally arriving at the end of the first chapter, I forget the content toward the beginning. Having nothing else I can do, I review the chapter from the start again. Then the entire structure of the chapter begins to bother me. I think things like, it seems better to take care of Theorem Seven before proving Theorem Three. So I create another notebook where I reorganize the whole chapter. I finally feel like I understand the first chapter, and I feel at peace, but it's troublesome that it took so terribly long. To get to the last chapter of a book with hundreds of pages is near impossible. I would very much appreciate it if somebody could teach me a quick way to read mathematical texts.
—Kodaira (translated by Hiro Lee Tanaka)
So we shall now explain how to read the book. The right way is to put it on your desk in the day, below your pillow at night, devoting yourself to the reading, and solving the exercises till you know it by heart. Unfortunately, I suspect the reader is looking for advice on how not to read, i.e. what to skip, and even better, how to read only some isolated highlights.
—Saharon Shelah
Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? Where does the proof use the hypothesis?
—Paul Halmos
Abundance of books makes men less studious.
—Hieronimo Squarciafico
If men learn [reading], it will implant forgetfulness in their souls; they will cease to exercise memory because they rely on that which is written, calling things to remembrance no longer from within themselves, but by means of external marks. What you have discovered is a recipe not for memory, but for reminder. And it is no true wisdom that you offer your disciples, but only its semblance, for by telling them of many things without teaching them you will make them seem to know much, while for the most part they know nothing, and as men filled, not with wisdom but with the conceit of wisdom, they will be a burden to their fellows.
—Socrates quoting Egyptian pharaoh Thamus in Plato's Phaedrus
You should know everything in this book, but don't read it!
—David Kazhdan