It is important to understand the different types of mathematical tools and their relationships. A broad generalization goes as follows:

• stuff (e.g. a set, or several sets, etc.)
• can be equipped with structure (e.g. functions, elements, relations, collections of subsets)
• that satisfy certain properties (e.g. equations, inequalities, inclusions)

As an example consider a function. A function is a pair of sets $X,Y$ equipped with a structure $f \subset X\times Y$ that satisfies $\forall x \in X \ \exists ! \ y \in Y \text{ s.t. } (x,y) \in f $.

• We can check properties: they are true or false.
• We can choose structures from a set of possibilities.
• We can choose stuff from a category of possibilities.

(Adapted from http://math.ucr.edu/home/baez/qg-spring2004/s04week01.pdf)

• One of the best student-friendly books to get familiar with many of the most important advanced tools used regularly in physics is Advanced mathematical methods for scientists and engineers by Bender and Orszag. “…a sleazy approximation that provides v good physical insight into what’s going on in some system is far more useful than an unintelligible exact result.”

• To get an overview of all the different math subfields this video: The Map of Mathematics is recommended.

• Great explanations of many advanced math topics, can be found in the "Infinite Napkin" book of Evan Chen.
• One of the best books to get familiar with many of the most important advanced tools is "Geometrical methods of mathematical physics" by Bernard F. Schutz
• A wonderful guided tour of the world of math and physics is "Road to Reality" by Penrose
• Also: Differential Geometry, Gauge Theories, and Gravity M. Göckeler, T. Schücker is a good read
• Further recommended texts to accompany these:
  1. Geometry of Physics, Frankel
  2. Intro to Lie Algebras & Rep. Th., Humphreys
  3. Geometry, Topology,& Physics, Nakahara (another useful survey)
  4. Spin Geometry, Lawson & Michelson

• Robert Geroch, Mathematical Physics
• Ivonne Choquet-Bruhat, Cecile DeWitt-Morette, and Margaret Dillard-Bleick, Analysis, Manifolds, and Physics
• Timothy Gowers, June Barrow-Green, Imre Leader "The Princeton companion to mathematics"

Mathematics and physics have gone their separate ways for nearly a century now and it is time for this to end.Topology, Geometry and Gauge Fields: Interactions by Naber

Physics doesn’t simply borrow the language from mathematics, but manipulates it, adapts it, and reinvents it according to the needs for a description of physical reality. The magic about physics is transforming the pure logic of mathematics into a beautiful narrative about nature. A poet couldn’t write poetry without a language, but language is not sufficient to make poetry. The same happens in science: mathematics is the language, but it takes physics to do poetry.

https://arxiv.org/pdf/1710.07663.pdf#page23