Those are good characterizations which capture more than you may know!, and provide a short/broad overview of the domain of set theory. But as with any field in math, how things look from the "outside in" isn't always quite as rich as you'll discover when you "enter into the study of and discourse within a domain:
The more you learn about set theory/theories,
the clearer the relationships you speak of will become,
the more readily you'll understand those relationships as a lens
through which you can view mathematics itself,
the more apparent it will be as to how useful understanding those relationships
can be when you see how set theory has served well as a foundation for much of modern math.
The more you learn about all the finer points of set theory, you'll find those patterns and relations to be even richer, more expansive, and more intricate than you first understood them to be.
You'll encounter some anomalies, as well. Perhaps you'll find some holes, and even incompatible theories: you'll find disagreement about foundational issues in set theory, and many points of departure, many of which have led to rich and varied resolutions, depending on the points of contention and of departure of various set theorists and logicians over time. But even these will reveal a pattern of sorts.
You'll also acquire the "language" of set theory, which will give you the vocabulary, the grammar, and the "tools" to better articulate the patterns and relationships you encounter. Sometimes, when we don't know a lot about a topic or a branch of math, as with any topic, it's hard to find the words to fully describe and capture the patterns we see. And it's hard to see what we don't have the words to describe.
For an overview of set theory
- This would be a great start: overview of Set Theory, with links you can click on to pursue certain threads or sub-topics of interest.
- See also the wikiBook on Set Theory.
These links will give you a taste of just how pervasive set theory is in terms of its "uptake" in mathematics, in general.
This is just a start at addressing your question. By no way is it a definitive, all-encompassing "answer". Personally, I enjoy finding patterns, and I also enjoy making connections across various domains of math, noticing patterns and common threads which weave their way through mathematics in ways that cannot be unraveled without destroying the very fabric of mathematics. These threads, too, form patterns.
Just some initial thoughts.