What pattern does set theory study? If mathematics is the science of patterns, what pattern does set theory study?
My thoughts so far: set theory studies the pattern of relationships between members of a collection and between collections.
Background: This isn't a homework question, I'm starting on Introduction to Mathematical Thinking and ensuring I have a good grounding in the fundamentals of set theory.
 A: I don't think that mathematics studies patterns. That's a huge oversimplification. It's like saying that physics studies things that move, or that historians study things that happened.
Allow me to preface the answer by pointing out that to a non-mathematician, mathematics is about solving equations with integrals and sines - despite the fact that it really isn't what mathematics is about. Similarly any current research in any advanced field is not seen by the naked eye. To say that set theory is about membership is roughly like saying that measure theory is about the length of intervals and their intersections or unions. It's not false, but as I pointed out above, it's a huge oversimplification, and it usually stems from not being familiar with what a set theorist (or a measure theorist, or a mathematician in general) does.
But let us take this oversimplification as it is right now. If set theory does study patterns of something, then I'd have to say that it studies patterns of provability and the lack thereof.
This of course is developed to patterns in models and universes of sets, and in some sense patterns of properties of these models, and properties of properties of these models, and so on.
For example, I want to show that statement $\varphi$ is not provable without the axiom of choice. I will begin with a specific model of set theory and try to build a universe where the axiom of choice and $\varphi$ fail, and then I will look for a model where $\varphi$ holds but the axiom of choice fails.
The next step is to analyze my proof and see which of the particular properties of the initial universe I used. Did I use this, or that, or neither? I will slowly clean up the dirt until I end up with the following result "If we start with a universe of set theory in which the axiom of choice, and an additional list of properties hold, then by doing this and that and that we can construct a universe of set theory in which $\varphi$ fails and the axiom of choice holds; and in taking a slightly different route we can make $\varphi$ true but the axiom of choice false.
That would be a pattern, so to speak. Patterns of provability, patterns of universes of set theory. I might analyze the technical methods of the construction and find the patterns there, and then I can try and come up with the following conclusion "If we use method A then we are guaranteed that B happens". This would be a pattern in the method of how we prove or construct new universes.
A: 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. 
A: Although technically all patterns can be reduced to patterns of set membership, this is rarely the best way to think about them.
I don't think that set theory should be characterized as the study of a single type of pattern, although there are sub-fields that can be approximately characterized in this way.
Here are some examples of patterns studied in set theory, although their definitions may not be very accessible to the lay person.


*

*patterns of consistency strength (mostly just a linear structure, but a very important one)

*patterns of extenders (in inner model theory)

*patterns of scales (in descriptive set theory)

A: Sets are logic! You write down some property P that objects might have, and then there is* a set S that corresponding to P. The set 'captures' the property P in the sense that some object has property P if and only if that object is an element of S.
Conversely, if you have a set S, you can define the property "is an element of S". 
Because a set is a mathematical object, I can manipulate it and reason with it conveniently. Writing a function that takes a property as input seems strange. How would you react to seeing f(even)? But if I had a set S of things that are even f(S) doesn't seem all that strange.
*: there is a caveat: diagonalization causes problems. e.g the property is a set that doesn't contain itself doesn't have a corresponding set. There are a number of ways to deal with this issue.
