This answer may be viewed as a bit odd and out of fashion by some, but one direction in which you could go--which intersects with commutative algebra, algebraic number theory, and discrete mathematics--is factorization theory.
Back when you were memorizing your multiplication tables in elementary school, the Fundamental Theorem of Arithmetic made that relatively easy in that there was only one way to break down a positive integer (greater than 1) into primes (which can be thought of as "building blocks", in a sense).
The generalization of this property that is discussed in the standard undergraduate abstract algebra sequence is, of course, the concept of the unique factorization domain (UFD), also called a factorial domain. The canonical example of an integral domain that does not enjoy unique factorization is the following example in $\mathbb{Z}[\sqrt{-5}]=\{a+b\sqrt{-5}\,\vert\,a,b\in \mathbb{Z}\}$:
$6=2\cdot3 = (1-\sqrt{-5})(1+\sqrt{-5})$.
One can use the standard norm function for rings of algebraic integers to show that 2,3, and $1\pm \sqrt{-5}$ are all irreducible in $\mathbb{Z}[\sqrt{-5}]$, and that the two factorizations of 6 above are distinct.
What's interesting, though, is that it turns out that while nonunit elements of $\mathbb{Z}[\sqrt{-5}]$ may not factor uniquely into irreducibles, any two factorizations of a fixed nonzero nonunit element have the same number of irreducible factors. Such a domain is called a half factorial domain (or HFD). In a short, two page paper in the Proceedings of the AMS, L. Carlitz characterized all HFDs amongst rings of algebraic integers via the (ideal) class group. Such rings have unique factorization precisely when the class group is trivial and are HFDs precisely when the class group is isomorphic to $\mathbb{Z}_2$. So, in the context of rings of algebraic integers, the size of the class group gives a measurement of "how far" we are from unique factorization.
Since Carlitz's seminal 1960 paper, there has been a lot of work done in factorization theory (and yes, this was my area of research while I was in academia, so I'm not exactly unbiased). Currently, the only book on the topic is Geroldinger and Halter-Koch's Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory. Make no mistake, this book is an extremely dense read, but it's very thorough and has an excellent bibliography.
Other sources (some of which are a bit more accessible to the newcomer) include:
Chris Ferrie has a new series out that includes:
- Quantum Physics for Babies
- Newtonian Physics for Babies
- Optical Physics for Babies
- Quantum Entanglement for Babies
Although not specifically concerning mathematical content, they do include in passing some mathematical ontology. The Newtonian book, for instance, does mention force being the product of mass and acceleration and describes acceleration. Not the best answer to my question, but I still am not finding as much material as I know is out there.
These are a bit like Basher books for toddlers. Big letters with colorful minimalistic pictures. Very simple sentences that stress the big new words the pictures are showing. The same pedagogical goal of just making these words feel safe to the young ones so that they feel comfortable "seeing the map" without needing to understand on any deep level what any of the "places" are really about. Early connections and all.
In looking, I've also found work like "ε-Red Riding Hood and the Big Bad Bolzano-Weierstrass Theorem" by Sunshine DuBois and Colin MacDonald, which although is very cute and clever, is not of the level or content that my child can enjoy. I think there is a clear distinction between a simple board-book style story with a simple core that mentions the big words in meaningful but not essential ways versus stories that have the appearance of children's stories but deliver highly mathematical content that the story depends upon for meaning. I know of many examples of the latter, but they clearly don't have the same pedagogical value.
Best Answer
Your question is philosophical rather than mathematical.
A colleague of mine told me the following metaphor / illustration once when I was a bachelor student and he did his PhD. And since now some years have passed I can relate.
It is hard to write it. Think about drawing a huge circle in the air, zooming in, and then drawing a huge circle again.
This is all knowledge:
All knowledge contains a lot, and math is only a tiny part in it - marked with the cross:
Mathematical research is divided into many topics. Algebra, number theory, and many others, but also numerical mathematics. That is this tiny part here:
Numerical Math is divided into several topics as well, like ODE numerics, optimisation etc. And one of them is FEM-Theory for PDEs.
And that is the part of knowledge, where I feel comfortable saying "I know a bit more than most other people in the world".
Now after some years, I would extend that illustration one more step: My knowledge in that part rather looks like
I still only know "a bit" about it, most of it I don't know, and most of what I had learned is already forgotten.
(Actually FEM-Theory is still a huge topic, that contains e.g. different kinds of PDEs [elliptic, parabolic, hyperbolic, other]. So you could do the "zooming" several times more.)
Another small wisdom is: Someone who finished school thinks he knows everything. Once he gained his masters degree, he knows that he knows nothing. And after the PhD he knows that everyone around him knows nothing as well.
Asking about your focus: IMO use the first few years to explore topics in math to find out what you like. Then go deeper - if you found what you like.
Are there "must know" topics? There are basics that you learn in the first few terms. Without them it is hard to "speak" and "do" math. You will learn the tools that you need to dig deeper. After that feel free to enjoy math :)
If your research focus is for example on PDE numerics (as mine is) but you also like pure math - go ahead and take a lecture. Will it help you? Maybe, maybe not. But for sure you had fun gaining knowledge, and that is what counts.
Don't think too much about what lectures to attend. Everything will turn out all right. I think most mathematicians will agree with that statement.