This is my (current) reading recommendation list for math.
General advice for beginners: How to Read Math, Calculus Made Easy. Several students have told me that they enjoyed the Art of Problem Solving books, and the book "Proofs: A Long-Form Mathematics Textbook" by Jay Cummings looks like an entertaining introduction to writing proofs. Gilbert Strang's "Introduction to Linear Algebra" is widely regarded as a great beginning textbook for linear algebra. I've heard good things about Spivak's "Calculus" as an introduction to more rigorous mathematics, although it may be somewhat difficult. The youtube videos by Mathologer, 3Blue1Brown, Vi Hart, and Numberphile are fun. Also, Wikipedia is surprisingly helpful for self-studying math if you learn to skip past the bits that go over your head and come back to them later, and searching Google for "Notes on [insert subject here]" usually turns up a decent exposition of any undergraduate or graduate level subject.
For highschool (or older) students who want to get good at competitive/fundamental math:
Number Theory: LeVeque's book "Fundamentals of Number Theory", Baker's "A Concise Introduction to the Theory of Numbers", "250 Problems in Elementary Number Theory" by Sierpinski, Mordell's "Diophantine Equations" (out of print, but it's on Library Genesis). A nice organizing principle for the foundations of number theory is the Euclidean algorithm and its generalizations (Bezout's identity, unique factorization for e.g. the Gaussian integers and Z[√2] and Z[x], continued fractions and the Pell equation, computation of quadratic residues, the arithmetic of ideals, infinite descent and "Vieta jumping", etc.) - when I teach number theory I usually try to hammer on this theme. Also, it is healthy to know what Peano's axioms are and how all of number theory can be derived from just that tiny list of axioms.
Combinatorics/CS: Sedgewick's "Algorithms in C" or "Introduction to Algorithms" by CLRS, "Principles and Techniques in Combinatorics", "Problem-Solving Strategies" by Engel, "A Course in Combinatorics" by van Lint and Wilson, "Enumerative Combinatorics" by Richard Stanley, maybe also "Problems from the Book". Also, try working your way through the USACO training pages (here). Even if you get very good at combinatorics, there will always be some problems you can only solve with luck, inspiration, or ugly casework (to understand why this happens, it helps to study the famous "P vs NP" problem from theoretical computer science). Nevertheless, there are portions of combinatorics which can be solved reliably and mechanically, and understanding these portions will help you solve the hard and ugly problems: mostly, these portions end up coming from algebra, so to get better at combinatorics you should also study algebra!
Geometry: "Geometry Revisited" by Greitzer and Coxeter, Prasolov's "Problems in Plane and Solid Geometry" (available online here), "Episodes in 19th and 20th Century Euclidean Geometry" by Honsberger, my own notes on cross ratios (work in progress, here), "Adventitious Quadrangles: A Geometrical Approach" by Rigby, maybe Evan Chen's handout on directed angles and his book "Euclidean Geometry in Mathematical Olympiads". If you are curious about what a rigorous axiomatic foundation for geometry looks like, check out Hilbert's book "The Foundations of Geometry" (available online here thanks to Project Gutenberg), as well as Tarski's axioms for elementary geometry.
More geometry advice: Do lots of locus problems to build up your geometric intuition (the goal is to be able to answer the question: "if one of these points in this diagram moves, then how will the other points move in response?"). Avoid the cheap tricks (such as converting everything to algebra, overuse of trigonometry, "complex numbers", "barycentric coordinates", "apply a projective transformation to send a line to infinity", "invert the diagram to turn circles into lines", "it's enough to check the statement holds at three points", etc.), try to learn the underlying philosophy and intuition behind geometry properly. Never apply a symmetry to an entire diagram, the correct application of symmetry relates points to other points on the same diagram (good examples: the Fermat point via a 60 degree rotation, Fagnano's problem via reflections, the Euler line via homothety, most applications of spiral similarity, perspectivity/cross ratio chase arguments such as the proof of Pappus's Theorem or Desargues's Theorem; nonexample: the wrongheaded way most people try to apply inversion).
Algebra and Inequalities: "A < B" by Kiran Kedlaya (online notes), Mildorf's notes on inequalities, "Inequalities" by Hardy, Littlewood, and Polya, "Positive Polynomials: From Hilbert's 17th Problem to Real Algebra", Khovanskii's "Fewnomials" (just the first few chapters), learn how to check if a quadratic form is positive definite (my own notes on this are here), learn about Fourier-Motzkin elimination and Farkas' Lemma about systems of linear inequalities, make sure you understand the fundamental theorem of symmetric polynomials, read about Sturm Chains, Resultants, Grobner Bases, and Cylindrical Algebraic Decomposition from Wikipedia or elsewhere, a few chapters of Dummit & Foote's book on algebra (maybe "A Book of Abstract Algebra" by Pinter first), maybe "The Cauchy Schwarz Master Class", "The Entirely Mixing Variables Method" by Phan Kim Hung, and these slides by Curtis Greene. Keep in mind that algebra is the most "learnable" subfield of math, plus it is generally useful in every other area of math, so time spent mastering it is never wasted.
Advice on getting started with writing proofs: in my opinion, before learning the rules of formal logic, it is best to read books and papers written by a variety of real mathematicians (not just textbooks - check out some of the books and papers below) and to try to imitate the way that they write their proofs. Just be aware that they might leave out many steps (in order to make the text shorter or to focus on the important details of the argument) and occasionally make mistakes - don't mindlessly mimic the shortcuts. Keep in mind that the level of detail in your proofs should depend on the difficulty of the problem you are solving: for difficult problems it is ok to skip some easy steps, but for easy problems you should go through those steps in detail. Be aware that geometry is a subject where many details (particularly "configuration issues") are often swept under the rug, despite geometry's reputation as the right setting for learning the rules of axiomatic proofs - in fact, I believe that the best settings for learning how to prove things rigorously are number theory (starting from the Peano axioms) and functional equations (which usually require meticulous attention to the logical details). Spend some time trying to understand the fundamental role that induction plays in number theory and that least upper bounds (aka supremums) play in the theory of the real numbers. Also get some practice with discovering bijections in combinatorics, and learn about how bijections are used in set theory to prove things about the cardinalities of infinite sets. Maybe check out the textbook "Proof and the Art of Mathematics" by Joel David Hamkins for an introduction to informal-but-correct proofs. [When you feel ready to learn the rules of formal logic, maybe start with some form of propositional logic (I am fond of the extremely simple Resolution proof system which is used in SAT-solving, but most textbooks cover Natural Deduction proof systems instead), then progress to first-order logic - make sure you at least know the statement of Gödel's Completeness Theorem, even if you don't know its proof (the proof of the completeness theorem for first-order logic is actually very insightful, so do make sure that you learn it eventually). However, formal logic is a topic which is hard to appreciate when you are young - this is why most of my book recommendations for logic are aimed at adults.]
For advanced highschool students, undergraduates, or older students who want to learn something interesting:
General math: "A=B" by Marko Petkovsek, Herbert Wilf and Doron Zeilberger (online here), "100 Great Problems of Elementary Mathematics" by Heinrich Dorrie, "Linear Algebra Done Right" by Axler (online here) and maybe "Linear Algebra Done Wrong" by Treil (online here), "Divergent Series" by Hardy, "Determinantal Identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley" by Brualdi and Schneider, "Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra" by Cox, Little, and O'Shea, "Visual Complex Analysis" by Needham, "A Singular Mathematical Promenade" by Étienne Ghys (online here), "Tsinghua Lectures on Hypergeometric Functions" by Heckman (online here), "Elementary Calculus: An Infinitesimal Approach" and "Foundations of Infinitesimal Calculus" by Keisler (online here and here).
Graphs: "Graph Theory" by Diestel, "Four Colors Suffice" by Robin Wilson, "Planar Graphs: Theory and Algorithms" by Nishizeki and Chiba, "Expander Graphs and Their Applications" by Hoory, Linial, and Wigderson, "On the Shannon Capacity of a Graph" by Lovasz, these notes on spectral graph theory by Spielman.
Topology: "Algebraic Topology" by Hatcher (online here), "Calculus on Manifolds" by Spivak, "Geometry and Topology of Three-Manifolds" by Thurston (here), "Differential forms in algebraic topology" by Bott and Tu, "Topology" by Munkres.
Fun: "Winning Ways for your Mathematical Plays" by Conway, Guy, and Berlekamp (also "On Numbers and Games" by Conway, the online collections "Games of No Chance" and "More Games of No Chance" and Vol 3, Vol 4, and Vol 5, "Dots and Boxes: Sophisticated Childs Play" by Berlekamp, and "Combinatorial Game Theory" by Aaron Siegel), anything by Martin Gardner, "Tracking the Dovetail Shuffle to its Lair" by Bayer and Diaconis, "Division by Three" by Doyle and Conway (also, the followup paper "Division by Four"), "Fractals, Chaos, and Power Laws", "One Hundred Prisoners and a Lightbulb", for the Angel problem start with Conway's description of the problem and this thesis by Kutz and then check out "A Solution to the Angel Problem" by Kloster (there are several other solutions out there as well), "Scheduling Algorithms for Procrastinators" (online here), "A Knowledge-based Approach of Connect-Four - The Game is Solved: White Wins" by Victor Allis.
More Number Theory: "Multiplicative Number Theory" by Davenport, "Number Theory" by Borevich and Shafarevich, "Arithmetic of Elliptic Curves" by Silverman, "Number Theory" by Andrews, "A Course in Arithmetic" by Serre, "Solving the Pell Equation" by Lenstra, Jr.
Physics-related stuff: "The Course of Theoretical Physics" by Landau and Lifshitz, some introductory quantum notes by Watrous (or these notes from Aaronson, or "Quantum Computation and Quantum Information" by Nielsen and Chuang), Coleman's Quantum Mechanics in Your Face, McQuarrie's "Physical Chemistry: A Molecular Approach", "QED: The Strange Theory of Light and Matter" by Feynman, and Earman's "A Primer on Determinism". For some generalized probability theory see "Probabilistic theories with purification" and "Informational derivation of quantum theory" or the book "Quantum Theory from First Principles: An Informational Approach" by Chiribella, D'Ariano, and Perinotti (also, "Nonlocality beyond quantum mechanics" by Popescu, "Quantum Theory From Five Reasonable Axioms" by Lucien Hardy, and maybe "Quantum Theory and Beyond: Is Entanglement Special?" by Dakic and Brukner). The ZX Calculus is fun to learn as well.
Philosophy and practice of mathematics: Jacques Hadamard's essay on "The Psychology of Invention in the Mathematical Field", Imre Lakatos's book "Proofs and Refutations", Penelope Maddy's papers "Believing the Axioms" (I and II), Benacerraf's "What Numbers Could Not Be" (maybe also "When is one thing equal to some other thing?" by Mazur), Eugenia Cheng's "Mathematics, morally", Tegmark's "The Mathematical Universe", "Alternative Set Theories" by Holmes, Forster, and Libert (maybe also "Foundations of Mathematics in Polymorphic Type Theory" by Holmes), "Exploring categorical structuralism" by Colin McLarty, "Non-Well-Founded Sets" by Aczel, Andrej Bauer's "Five Stages of Accepting Constructive Mathematics", Edward Nelson's "Radically Elementary Probability Theory" and "Internal set theory: A new approach to nonstandard analysis".
Other: "Convex Bodies: The Brunn-Minkowski Theory" by Schneider, "Algebraic Combinatorics on Words" (online here), "Integration in Finite Terms" by Rosenlicht, "Binomial Coefficients and Jacobi Sums" by Hudson and Williams (also, "Binomial Coefficients Modulo Prime Powers" by Granville), "A Simple Proof of Sharkovsky's Theorem" by Du, "Puzzles in geometry which I know and love" by Anton Petrunin, "Linear Orderings" by Rosenstein, and the Metamath website.
For adults:
Logic: "Model Theory" by Hodges or "Model Theory: An Introduction" by Marker, Hedman's "A First Course in Logic", "Handbook of Automated Reasoning", the study guide "Teach Yourself Logic" (online here), "Introduction to Metamathematics" by Kleene, "Fixing Frege" by Burgess, "Propositional Proof Systems, the Consistency of First Order Theories and the Complexity of Computations" by Krajicek and Pudlak, "Short Proofs are Narrow - Resolution made Simple" by Ben-Sasson and Wigderson and "A Combinatorial Characterization of Resolution Width" by Atserias and Dalmau, "DRAT and Propagation Redundancy Proofs Without New Variables" by Buss and Thapen, "Logical Foundations of Proof Complexity" by Cook and Nguyen, "Subsystems of Second-Order Arithmetic" by Simpson, "Hilbert's Tenth Problem is Unsolvable" by Davis, "A Survival Guide to Presburger Arithmetic" by Haase, some nice lecture notes on set theory for a quick intro or "Set Theory" by Jech for way more detail, "The Set-Theoretic Multiverse" and these slides by Hamkins have an interesting insight into how forcing really works, "The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings" by Kanamori, "Admissible Sets and Structures" by Barwise, "Determinacy of Infinitely Long Games" by Martin, "Model-Theoretic Logics" edited by Barwise and Feferman, for Gentzen's consistency result see "Basic Proof Theory" by Troelstra and Schwichtenberg (chapter 10) or "Proof Theory: An Introduction" by Pohlers, for modal logic see "The Logic of Provability" by Boolos, for lambda calculus I liked "Lambda Calculus: Some Models, Some Philosophy" by Dana Scott.
Groups: "Finite Group Theory" by Isaacs, "Finite Permutation Groups" by Wielandt, "Finite Groups: An Introduction" by Serre, "Representation Theory: a First Course" by Fulton and Harris, "Fuchsian Groups" by Svetlana Katok, "Trees" by Serre, "Transitivity of Permutation Groups on Unordered Sets" by Cameron, "Generic Polynomials - Constructive Aspects of the Inverse Galois Problem" by Jensen, Ledet, Yui.
Algebra: "Commutative Algebra with a View Toward Algebraic Geometry" by Eisenbud, maybe "Commutative Algebra" by Matsumura (make sure to get the 2nd edition) or "Introduction to Commutative Algebra" by Atiyah and MacDonald if you are pressed for time, "Newton Polyhedra, A New Formula for Mixed Volume, Product of Roots of a System of Equations" by Khovanskii, "Maximally Complete Fields" by Poonen, "The Structure of Finite Algebras" by Hobby and McKenzie, "Matroid Theory" by Welsh (also "Matroid Theory" by Oxley and "Lectures on Matroids" by Tutte). I personally liked Lang's "Algebra".
Analysis: "Principles of Harmonic Analysis" by Deitmar and Echterhoff and "Fourier Analysis on Number Fields" by Ramakrishnan and Valenza, "The Laplace Transform" by Widder, "Real Analysis: Modern Techniques and Their Applications" by Folland, "Integrals of Nonlinear Equations of Evolution and Solitary Waves" by Peter Lax.
Homotopy/Homological Algebra: "Characteristic Classes" by Milnor and Stasheff, "Morse Theory" by Milnor, "Model Categories" by Hovey, "An Introduction to Homological Algebra" by Weibel, "Spectral Sequences: Friend or Foe?" by Vakil.
Algebraic Geometry: Ravi Vakil's online notes (here), "Principles of Algebraic Geometry" by Griffiths and Harris, "Advanced Topics in the Arithmetic of Elliptic Curves" by Silverman, "Heights in Diophantine Geometry" by Bombieri and Gubler, the Stacks project (online here), "Neron Models" by Bosch, "Stable n-pointed trees of projective lines". I personally got a lot out of "Introduction to Etale Cohomology" by Gunter Tamme, and ended up writing my own notes on cohomology based on it, but that might just be a consequence of that book being my first introduction to the subject.
Even More Number Theory: "Lectures on Sieves" by Atle Selberg (in vol II of his collected works), "Transcendental Number Theory" by Baker, "Algebraic Number Theory" by Cassels and Frohlich (especially Chapters VI and VII), "Class Field Theory" by Childress, Milne's notes, "Primes of the Form x2 + ny2" by Cox, "Local Fields" by Serre, "Algebraic Groups and Number Theory" by Platonov and Rapinchuk, "Deforming Galois Representations" by Mazur or "Deformations of Galois Representations" by Gouvea, "p-adic Numbers, p-adic Analysis, and Zeta-Functions" by Koblitz, "Notes on Weil 2" by Katz.
Modular Forms: "Automorphic Forms and Representations" by Bump, "A First Course in Modular Forms" by Diamond and Shurman, "Notes on Jacquet-Langlands' theory" by Godement (here).
Algorithms: "Approximation Algorithms" by Vazirani, "Mathematics and Computation" by Wigderson (online here), "Probabilistic Graphical Models: Principles and Techniques" by Koller and Friedman (also "Causality: Models, Reasoning, and Inference" by Pearl), "Foundations of Cryptography" by Goldreich, "The Multiplicative Weights Update Method: a Meta Algorithm and Applications" (also "Regularity, Boosting, and Efficiently Simulating Every High-Entropy Distribution" and "Regret Analysis of Stochastic and Nonstochastic Multi-armed Bandit Problems"), "Introduction to Numerical Algebraic Geometry", "Learnability and the Vapnik-Chervonenkis Dimension", "Factoring Polynomials with Rational Coefficients" by LLL.
Other: "The Wild World of 4-Manifolds", "Probability on Graphs" by Grimmett, "Additive Combinatorics" by Rusza, "Mathematics Made Difficult" by Linderholm, "What Is a Quantum Field Theory?" by Talagrand and "Quantum Field Theory in a Nutshell" by Zee and "The Quantum Theory of Fields" by Steven Weinberg, "User's guide to viscosity solutions of second order partial differential equations", "All Those Ramsey Classes" by Hubicka and Nesetril (and "Graham-Rothschild Parameter Sets" by Promel and Voigt).