Recommendations

Here's some books, videos, and course notes I'd recommend. The lists generally go from introductory to advanced in order, and more advanced topics are later in each subject list. Freely available resources are linked.

General:

Garrity's All the Mathematics You Missed: Exactly what it says, a good overview of everything that (should) be covered in undergrad.

Stewart's Concepts of Modern Mathematics: Want to know why someone would actually chose to study maths? Here's a good start point.

Velleman's How to Prove It: A really nice look at methods of proof, set theory, and logic.

Algebra:

As a mathematician you really can't get away with avoiding algebra, no matter where you go. Thankfully it's also a beautiful subject, when given the respect it deserves.

Linear Algebra

Most introductory linear algebra texts are pretty good. I still haven't developed a solid preference, but as far as I'm concerned a first course should be very computational and hands on.

3Blue1Brown's Essence of Linear Algebra series: This is how I actually learned basic linear algebra, should be bundled with any first course.

Second course: Sheldon Axler's Linear Algebra Done Right. This is the perfect second look at linear algebra to me. The rest of abstract algebra starts to look a lot less distant to linear when everything is viewed through mappings. A lot of the weirder seeming basis considerations in an intro course will start to make sense here too.

Really advanced: You can either consider learning more about modules and fields or learning tensor algebra as the next step, both these paths converge eventually. Both are better learned for their own reasons though, in my opinion.

Group Theory

Group theory was my hook into math, a lot of the classic books are pretty dated, but you can't go too wrong, especially if you're willing to work through some things on paper and make some shapes from cardboard. A key insight in learning group theory is that it's very combinatorial - we have really good control of groups by counting how many of a given *thing* there is, where that thing can be subgroups, orbits, etc. Here's some favourites.

This thread: This should honestly be everyone's first encounter with groups. Perfectly paced, exploratory, and hits most major topics.

Grossman's Groups and Their Graphs: this is hard to get a hold of and a very non-standard course, but I can't not include it, it was my first maths book.

Dummit & Foote's Abstract Algebra part 1: There's more than you want for a first course here, but you won't find a better set of examples. Be willing to skip things and go back.

Abstract Algebra (First Encounter)

Once you've got a strong base in group theory it's time to look at what happens when we add or take away structure. This is the gateway to most "fun" topics, but is a whole lot of fun on its own.

Dummit & Foote's Abstract Algebra: One of the only reference books that's also an excellent place to learn. Has most of undergraduate (and a fair chunk of graduate) algebra in it. Once you've covered a fair chunk of part 1 and 2 you can start hopping around the book a bit more freely. Particular highlights if you're looking for advanced topics are Galois theory, commutative algebra, and representation theory.

Aluffi's Algebra Chapter 0: This is the book that finally convinced me categories were fun. I personally think it's better to go in after a first course in abstract algebra and flick through, working through any categorical constructions that come up. It will change the way you see algebra.

Graph Theory

I think skipping graph theory at undergrad is a massive misstep. It's probably the easiest way to access genuine research mathematics and is generally a really nice subject to give the flavour of math.

Trudeau's Graph Theory: Really nice introduction, cheap, short, and fun.

Harris' Combinatorics and Graph Theory: A really nice book covering anything you would need to know in graph theory. Gives lots of motivation for research problems too.

Category Theory

Aluffi's Algebra Chapter 0: As above, this is the book that finally convinced me categories were good. Grab a pen and paper and work through the various constructions.

Riehl's Category Theory in Context: The new canonical reference. Filled with examples and excellent explanations of dense topics.

Bradley's Topology: A Categorical Approach: A favourite of mine. The goal of this book is to teach a second course in topology as an excuse to teach categories, effectively doing what Aluffi did for categories in algebra.

Tai-Danae Bradley's blog Math3ma: A set of really nice blog posts about category related things and teaching fundamentals.

Analysis:

Analysis is often the first hurdle in actually getting through a math degree, but the key to so much modern math, and eventually a really interesting field on its own.

Calculus and Real Analysis

Real analysis is usually viewed as a stepping stone into proper analysis, I really don't agree with this but I'm not sure what the solution is.

Stewart's Calculus is my preferred first book, computational without feeling too much like a science book, I think it should be read alongside Spivak's book though, even if Spivak is only used for reference.

For a better look at analysis without the computational focus I like Bartle, but I haven't went back to it in a few years.

Complex Analysis

Needham's Visual Complex Analysis: This book is mandatory for anyone in any way interested in complex analysis, which is probably most mathematicians. Highlights the rigidity and geometric nature of complex analysis throughout.

Ahlfors' Complex Analysis: The classic reference. Short and clever. Best used after a first course.

Differential Equations:

I'm really not sure what the best first course is, I largely used Paul's online notes and learned from lectures.

Evans' Partial Differential Equations: A very heavy book, but well written with lots of nice tricks.

Olver's Applications of Lie Groups to Differential Equations: A great book for those that never "clicked" with DEs. Builds up a theory to highlight the visual and algebraic structure of DEs.

Topology

Munkres' Topology: Good if you're either willing to work through the set theory chapter at the start, but I think it's a bit much.

Mendelson's Introduction to Topology: A really nice intro to metric spaces and topology, with the added benefit of being cheap.

Sutherland's Metric Spaces: A bit more dry and traditional than Mendelson, but short and very useful.

Lee's Introduction to Topological Manifolds: I have no experience with this particular book, but I have read a Lee's other two manifolds books. Given by their quality I doubt there's a better place to learn topology.

Hatcher's Algebraic Topology: This book deserves every bit of reputation it has. It's also completely free on his website, along with his own list of cheap books in algebra. Expect to read this book over and over again if you plan to go into algebraic topology, though that doesn't mean it's a bad place to start - pace yourself and be ready to skip over problematic sections and examples.

Curves and Surfaces, Manifolds

Week's The Shape of Space: Invaluable to anyone planning to go into topology or geometry. Pairs really nicely with any other book here.

DoCarmo: This is the standard here, for good reason, but it can be hard to find what you want with the size of it.

Milnor's Topology from the Differential Viewpoint: this book definitely needs a bit of topology going into it, but it's a great, gentle introduction to differential topology.

Lee's Introduction to Smooth Manifolds: Everything you need to know about smooth manifolds. I strongly prefer the first edition. Not a good first course in my opinion, but running straight from Milnor to here is a good transition.

Geometry:

I, and many others, are guilty of putting geometry courses under a different heading, what I have down as topology and manifolds above could rightfully be down here, drawing lines between fields in mathematics is dangerous.

Algebraic Geometry

I actually wrote an entire thread on the many issues with learning algebraic geometry, here's the main recommendations from there, in order:

Gathmann's plane algebraic curves

Smith's Invitation to Algebraic Geometry

Gathmann's algebraic geometry

Mechanics:

Although I am technically interested in all of physics, the only fields I feel in any way qualified to give recommendations for are mechanics and relativity. I strongly believe every mathematician should learn some mechanics, hopefully one of these recommendations will show you why.

Kleppner & Kolenkow's Mechanics: I still really like the examples and presentation in this book. One point where I think it fails is in giving a justification and broad picture of the study of mechanics. I'm writing some notes on the first chapters of the book aiming to fix this, find the current version here. Goldstein is the usual alternative, but I have no experience with it.

Arnold Mathematical Methods of Classical Mechanics: A really wonderful book, the perfect place for a mathematician to learn mechanics.

Dray's Geometry of Special Relativity: A very different approach to relativity, but will give you a new and very useful perspective.