This is my (current) reading recommendation list, inspired by emails I've received asking for reading suggestions.
General advice for beginners: How to Read Math, Calculus Made Easy. 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. The youtube videos by Mathologer, 3Blue1Brown, and Vi Hart are also decent. (I've also heard good things about Spivak's "Calculus" as an introduction to more rigorous mathematics, although it may be somewhat difficult.)
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", Mordell's "Diophantine Equations" (out of print, but it's on Library Genesis), "250 Problems in Elementary Number Theory" by Sierpinski.
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, the USACO training pages (here).
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.
More geometry advice: Do lots of locus problems to build up your geometric intuition. 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 (examples: the Fermat point, Fagnano's problem, the Euler line, most applications of spiral similarity, perspectivity/cross ratio chase arguments such as the proof of Pappus's Theorem or Desargues's Theorem; nonexample: the way most people try to apply inversion).
Algebra and Inequalities: "A < B" by Kiran Kedlaya (online notes), Mildorf's notes on inequalities, "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, read about Sturm Chains, Resultants, and Grobner Bases from Wikipedia or elsewhere, a few chapters of Dummit & Foote's book on algebra (maybe "A Book of Abstract Algebra" by Pinter), 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.
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, "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).
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, "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", "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", "A Solution to the Angel Problem" by Kloster (and as background, Conway's description of the problem and this thesis by Kutz, also there are several other solutions out there), "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 "Quantum Theory and Beyond: Is Entanglement Special?" by Dakic and Brukner). The ZX Calculus is fun to learn as well (maybe start here).
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, "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, "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 Higher Infinite: Large Cardinals in Set Theory from Their Beginnings" by Kanamori, "Admissible Sets and Structures" by Barwise, "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, "Commutative Algebra" by Matsumura (make sure to get the 2nd edition), "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). Also, 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, "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 I don't know if my experience generalizes.
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, "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).