In Teach Yourself Logic, Peter Smith provides a road map to Logic clearly indicating the stages one has to pass through in order to increase gradually one's knowledge in this area. This progression is explained in terms of concepts and of theorems being the entrance door to each stage.
What would a comparable road map to set theory look like?
How could be described , in terms of concepts, results, and directing problems at each stage an ascent from
(1) "baby" set theory
to
(2) intermediate set theory
and finally to
(3) higher set theory, or set theory proper
Of course what I am asking is rather difficukt to offer ( if not impossible), for, in a sense, one has to already know set theory to understand such a road map, but maybe an attempt can be made.
Remark: One related question I ask myself is : to which level belongs the construction of the various sets of numbers? Is it for set theory a goal, or a starting point?
Remark. In comments has been given a reference related to the question I ask here: Asaf Karagila, The five WH's of Set Theory. ( At : Website of the European Set Theory Society)
I am by no means an expert in set theory but I've seen a bunch of stuff relating to it, so here's my guess. You're welcome to edit it with more info if you have some.
(1) Baby set theory : that's the set theory everyone knows from the very beginning of their mathematical studies, the one that's needed for basically any mathematician :
having an understanding of the concept of set, knowing basic sets, for instance number sets as you mentioned, and knowing the constructions one can perform with sets : (ordered) pairs, cartesian product, (disjoint) unions, power sets, set of functions, subset of elements satisfying a certain property, etc. ; and understanding how to represent basic ideas prevalent in mathematics in terms of sets (what does "a set equipped with a binary operation" mean ?).
This also contains the basic ideas about "size" of sets: what is an injection, a surjection, a bijection, and what do these correspond to, intuitively ? Once you have all that, you're ready to move forward to something more interesting.
I don't know if this goes here or in "intermediate", but it's certainly popular enough that most mathematicians know it: you get acquainted with your first diagonal arguments, either Russell's "paradox", or Cantor's theorem (or both)
(2) Intermediate set theory : that's the set theory you get if you take an introductory class on logic and set theory at an undergraduate level. At this point the points of the previous section should all be reflexes, that you don't have to think about.
Here you get introduced to some more subtle ideas about size for instance, like ordinals, cardinals, their basic properties and surrounding objects. You also get acquainted with filters for instance.
This is also the point where you start worrying about the axioms, and so you probably learn a bit about model theory too, to get a handle on said axioms. You need to find out what axioms you use, and which ones are relevant to what you're doing : does this proof use the axiom of choice ? the axiom of foundation ? What's the continuum hypothesis?
You also learn about cardinal arithmetic ["computing sizes of sets"]; and basic cardinal arithmetic becomes sort of a game and you can answer questions like : are there as many sequences of reals as there are reals ? How many continuous functions $\mathbb{R\to R}$ are there ?
You also start noticing some subtleties in more advanced cardinal arithmetic and studying things like cofinality. You can start using cofinality in cardinal arithmetic in basic examples.
At this point, there are many examples of natural, easy to state ("easy" with the background mentioned above) questions that pop up, that you can't find the answer to on your own, and perhaps no one can (whether it's because it's independent from the axioms of set theory, or it's because no one knows if it is)
As Alex Kruckman pointed out in the comments, there is at this point a bit of ambiguity regarding what counts as "intermediate" and what counts as "advanced" : he mentions inner models and forcing, which are basic techniques in proving independence of statements : they're reflexes for set theorists, and stuff like the constructible universe $L$ or $HOD$, or forcing $\neg$ CH are routine to them, but undergraduate courses mostly end a tad too early to study them properly, so they're at the boundary between the two "levels".
(3) Advanced set theory : you've mastered all that was mentioned before, and you start doing way more subtle things.
At this point, cardinal arithmetic becomes harder (because the basic examples are trivial, and the ones you are faced with are on another level) and relates to cofinality, perhaps game theory, and the axioms you're using play a definite role. You start seeing statements that involve numerous "nonstandard" axioms (by which I mean : not present in the usual ZF(C) list), like "if Martin's axiom holds and GCH fails at such and such cardinals, then bla" [I have no idea if this pretend-example would make sense, I don't know what Martin's hypothesis is] , and you can prove independence results : "if ZF+ DC is consistent, then so is ZF+ godknowswhat".
The large cardinals and their gang start appearing, you have hypotheses like "$0^\#$ exists", or "there are at least $2^{\aleph_0}$-many Mahlo-cardinals" etc. etc.
I have gotten the impression (set theorists out there, correct me if I'm wrong) that a lot of modern set theory (so, set theory that fits into this "advanced set theory" bit of my answer) is about large cardinals; forcing axioms (forcing is a super interesting tool discovered by Cohen in the 60's that can be used very efficiently to prove independence statements, such as the very famous "if ZFC is consistent, it does not prove the continuum hypothesis"); more generally independence statements; and infinite combinatorics (which I take to mean cardinal arithmetic but also things like Ramsey theory)
What's very interesting is that all of these are deeply connected : studying Ramsey theory for instance naturally involves various large cardinals; and comparing different large cardinals involves studying independence of various statements etc. etc.