Here are my preferred/recommended books for self-study, teaching, etc.
College Algebra
College Algebra - OpenSTAX (This is what I follow for teaching.)
College Algebra - Michael Sullivan (Used for nice motivating real-world applications for students.)
Algebra - Gelfand & Shen (for honors algebra)
Elementary Geometry
Geometry - Ray, Richard, John (main textbook for teaching)
Elementary Geometry - Alexander & Koeberlein (for teaching materials & problems)
Trigonometry
Trigonometry - Lial, Hornsby, Schneider, Daniels (main textbook for teaching, contains great application problems)
Trigonometry - Gelfand & Saul (for honors trig)
Precalculus
Precalculus - Sheldon Axler (main textbook for teaching)
Precalculus - Michael Sullivan (for application purposes)
Single-Variable Calculus
Calculus Early Transcendentals - James Stewart (Absolute MUST)
Calculus - Michael Sullivan (for application problems for students)
Calculus - Varberg, Purcell, Rigdon (for conceptual review problems)
Calculus - Michael Spivak (for competitive practice, just ignore the proofs or formality)
Multivariable Calculus
Calculus Early Transcendentals - James Stewart (Absolute MUST)
Linear Algebra & Multivariable Calculus - Evan Chen
Differential Equations
Differential Equations & Linear Algebra - Edwards, Penney, Calvis (main textbook for teaching)
Linear Algebra
Elementary Linear Algebra - Spence, Friedberg, Insel (for teaching intro linear algebra for all STEM majors, contains beginner proof problems as well)
No Bullshit Guide to Linear Algebra - Ivan Savov
Intro to Math Proofs
Mathematical Proofs - Chartrand, Polimeni, Zhang (main textbook for teaching)
Book of Proof - Richard Hammack (another book use for teaching as well)
Proofs - Jay Cummings (entertaining for beginner math circles) Buy it Here!!
How to Read & Do Proofs - Daniel Solow
Discrete Math
Discrete Mathematics - Richard Johnsonbaugh (main textbook for teaching)
Discrete Mathematics and its Applications - Kenneth Rosen (for more application/real-world problems)
Mathematics, A Discrete Introduction - Edward (for advanced CS majors)
Mathematical Statistics
Set Theory/Symbolic Logic
Set Theory, A First Course - Cunningham (nice beginner's book, high school level/perfect for math circles)
Intro to Number Theory
Introduction to Number Theory - Mark Hunacek
Elementary Number Theory - James Strayer
Graph Theory
History of Mathematics
Problem Solving Mathematics (Intro to Competitive Math)
Mastering AMC8 & AMC 10/12 - Omega Learn (This is free and this is what I use to teach problem solving, including AoPS Prealgebra)
AoPS Prealgebra, Introduction to Algebra, Number Theory, Geometry, and Counting & Probability (This will help you coach math contests.)
Lecture Notes On Mathematical Olympiad Courses Series (this will change the way you problem solve, a bit advanced tho)
Abstract Algebra
Contemporary Abstract Algebra - Joseph Gallian (I use this for teaching beginner/computational concepts.)
Elements of Modern Algebra - Gilbert & Gilbert (I use this with Gallian, also amazing for computational based problems.)
Groups, Rings, and Fields - Wallace (I use this for teaching proof-based stuff and examples, perfect proof book for beginner undergrads)
Abstract Algebra - Herstein (I use this for practice proof-problems based on difficulty, very helpful reference)
Introduction to Abstract Algebra - Nicholson (With Wallace, I use this for rigorous proof concepts & problems for undergrads)
A Book of Abstract Algebra - Pinter (High School Level, Perfect for Math Circles)
How to Think About Abstract Algebra - Lara Alcock (for Math Circles)
A Friendly Introduction to Abstract Algebra - Matsuura (for Advanced Math Circles)
Abstract Algebra, an Introduction - Hungerford (beginner graduate level)
Algebra - Michael Artin (for advanced, elite, impatient learners who really likes to challenge themselves, this is Spivak in Algebra dimension)
Problems in Abstract Algebra - Wadsworth (brutal practice for PhD candidates)
Applied Abstract Algebra - Lidl & Pilz (for CS majors who took abstract algebra)
An Introduction to Abstract Algebra - Weintraub (A very neat honors level book,+ Field Theory, for olympiad/talented math circles like MIT Primes)
Real Analysis
Real Analysis w/ Proof Strategies - Cunningham (use for teaching, high school level book/motivating beginner undergrad book) Buy it Here!!
Introduction to Analysis - Wade (also use for teaching, more rigorous undergrad book)
Yet Another Intro to Analysis - Victor Bryant (high school level, funny & informal, great for understanding rigorous theorem in low level context)
Mathematical Analysis 1 & 2 - Canuto & Tabacco (has solutions for all exercises)
Introduction to Analysis - Arthur Mattuck (two sets of exercises for each chapter --> solutions for one set for each chapter)
How to Think About Real Analysis - Lara Alcock (for Math Circles)
Real Analysis - Jay Cummings (for advanced math circles) Buy it Here!!
Real Analysis - John M. Howie (best textbook for summer school)
Complex Analysis
Undergraduate Geometry for Math Teachers
Differential Geometry
Elementary Differential Geometry - Andrew Pressley (this is my teaching/learning material, pretty nice for undergrads)
Differential Geometry of Curves & Surfaces - Kristopher Tapp (I use this aside with Pressley, amazing visuals for undergrads)
Elementary Differential Geometry - Barrett O'Neill (more computational, perfect for Advanced Math Circles)
Differential Geometry of Plane Curves - Alencar, Santos, Neto (a bit more advanced but still motivating to read, all problem solutions in the back)
Differential Geometry of Manifolds - Stephen Lovett (motivating graduate level book or for advanced undergrads)
Partial Differential Equations
An Introduction to Partial Differential Equations with MATLAB (although I don't use it for Matlab, best introductory book, good for math circles)
Introduction to Partial Differential Equations - Peter J. Olver (more undergraduate/beginner-grad level, heavily applied than pure computations)
Topology
Topology without Tears - Morris (I use this book, WAY NICER TO READ compared to other books..., high school level/best for math circles)
An Introduction to Topology - Tej Bahadur Singh (more rigorous undergraduate book)
An Introduction to Topology Pure & Applied - Adams & Franzosa (LOTS OF VISUALS, more rigourous yet motivatingly readable)
REA's Topology - Milewski (I use this for nice beginner yet rigorous practice problems)
Metric Space Topology - Cheung (more rigorous practice problems for beginner graduates)
Introductory Topology Exercises & Solutions (more rigourous practice problems to add on Cheung)
Functional Analysis
Functional Analysis - Sergei Ovchinnikov (this is my teaching/learning material, nice for graduate beginners & adequately rigorous)
Introduction to Functional Analysis - Clason (motivating for advanced undergraduates)
Introductory Functional Analysis w/ Applications - Kreyszig (neat and introductory, but ancient ;_;)
Exercises in Functional Analysis - Costara & Popa (brutal practice for PhD students)
Representation Theory
Algebraic Geometry
Beginning in Algebraic Geometry - Ross & Clader (my teaching/learning material, nice for advanced math circles too)
Lie Algebra
Commutative Algebra
Elliptic Curves
Elliptic Curves - Lawrence C.
Fourier Analysis
Measure Theory
Measure, Integration, & Real Analysis - Sheldon Axler
Algebraic Number Theory
Algebraic Theory of Numbers - Pierre Samuel (Probably a great fresh start for grad students. Sadly its very ancient tho...)
Introductory Algebraic Number Theory - Alaca & Williams (This is the book I'd use to teach, but it is high school level, too easy for math circles)
Algebraic Number Theory - Chahal (This is what I follow to teach/learn graduate level.)
Algebraic Number Theory - Richard A. Mollin (Undergrad level but a bit more rigorous, probably perfect for a grad's beginner.)
A Conversational Introduction to Algebraic Number Theory - Paul Pollack (perfect book for math circles)
Algebraic Number Theory, A Computational Approach - William Stein (if you're an astrophysicist or cracked Berkeley CS major)
Algebraic Number Theory and Fermat's Last Theorem - Ian Stewart & David Tall (undergrad friendly book)
Problems in Algebraic Number Theory - Murty & Esmonde (a book to practice, especially for rusty upcoming PhD students)
Algebraic Topology
Algebraic Topology - Bray, Butscher, Rubinstein-Salzedo (High School Level Book, used for SUMaC)
Analytical Number Theory
Analytical Number Theory for Beginners - Pongsriiam (This is my teaching material.👌)
A Primer of Analytic Number Theory - Jeffrey Stopple
Problems in Analytical Number Theory - Murty (Amazing practice!!! Especially for upcoming PhD students)
Topics in the Theory of Numbers - Emil (become fully aware of all significant NT topics)
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Galois Theory
Galois Theory - Ian Stewart (nice introductory book, high school level/perfect for math circles)
Galois Theory - Tom Leinster (nice motivating book for advanced math circles)
Category Theory/Rings & Modules
Introduction to Rings & Modules - C. Musili (probably a nice introducing book for advanced undergrads/beginner graduates, nice for math circles)
Ring and their Modules - Paul E. Bland (nice graduate level book, includes Category Theory & Homological Algebra, for advanced math circles)
Differential Topology
A Short Course in Differential Topology - Dundas (neat graduate book)
Integral Equations
Integral Equations - Sharma & Goyal