Why study set theory?
We like to think that mathematics developed from the need of our ancients to count things. I have four sheep, you have sixteen camels, my tribe has ten dozens of men, you have six hundred wives... etc. etc. But if you look closely, counting how many things you have of a certain type, first required you have be able and collect them into one collection. The "collection of all sheep I have", or the "collection of men in my tribe", and so on.
Sets came to solve a similar problem. Sets are collections of mathematical objects which themselves are mathematical objects.
This, of course, doesn't mean that we should learn set theory just for that purpose alone. The applications of set theory are not immediate for finite collections, or rather sufficiently small collections. We don't need to think about pairs or sets with five elements as particular objects. Whatever we want to do with them we can pretty much do by hand.
Sets come into play when you want to talk about infinite sets. Infinite sets collect infinitely many objects into one collection. The set of natural numbers, the set of finite sets of sets of sets of natural numbers, the set of sets of sets of sets of sets of sets of irrational numbers, etc. Once you establish that mathematical objects can be collected into other mathematical objects you can start analyzing their structure.
But here comes the problem. Infinite sets defy our intuition, which comes from finite sets. The many paradoxes of infinity which include Galileo's paradox, Hilbert's Grand Hotel, and so on, are all paradoxes that come to portray the nature of infinity as counterintuitive to our physical intuition.
Studying set theory, even naively, is the technical spine of how to handle infinite sets. Since modern mathematics is concerned with many infinite sets, larger and smaller, it is a good idea to learn about infinite sets if one wishes to understand mathematical objects better.
And one can study, naively, a lot of set theory, especially under the tutelage of a good teacher that will actually teach axiomatic set theory in a naive guise. And this sort of learning can, and perhaps should, include discussions about the axiom of choice, about ordinals, and about cardinals. As Ittay said, and I'm agreeing ordinals and cardinals are two ways of counting, which extend beyond our intuitive understand that counting is done via the natural numbers, and allow us to count infinite objects.
If one couples these ideas with the basics of first-order logic, predicate calculus, and basic first-order logic, one understands how set theory can be used as a basis for modern mathematics. Which again, allows us to better see into some parts of mathematics.
Axiomatic set theory, on the other hand, is a mathematical field like any other. It has certain type of typical problems, and set theorists work in their typical or atypical ways to solve them, or at least understand them better. Axiomatic set theory does, however, handle the fine-grained problems that come from infinity better.
Why do I mean by that? A lot of the infinite sets in modern mathematics are countable or have size continuum. Rarely we run into larger sets (e.g. the set of all Lebesgue measurable sets is larger), but even then we rarely care about that. But now that we understand infinite sets better, we can ask questions like "Given an abelian group with such and such properties, is it necessarily free [abelian]?" usually we can prove these sort of theorems for countable objects, in this case countable groups, but not beyond that.
Sometimes we are interested in topology, which allows us to extend our control from countable objects to things that can be approximated "in a good way" with countable objects (like separable spaces). But even then we can ask questions which involved an arbitrary objects, and not necessarily one which has 'nice properties'.
It turns out that our lack of intuition for infinite sets is reflected in the lack of "naively provable structure" of infinite sets. We cannot even provably determine how many distinct cardinalities lie between the cardinality of $\Bbb N$ and $\Bbb R$. It might be none, or it might be one or two or many more. Here axiomatic set theory comes into play.
Axiomatic set theory deals with the additional axioms that we can require the set theoretic universe to have, and how they affect the structure of infinite sets. And this is the importance of set theory to mathematical research, as well. It deals with solving the existence or what sort of assumptions we need to prove, or disprove, the existence of certain objects.
These objects, while seemingly arbitrary, can have a great influence and strong effects on the structure of "mathematically interesting sets". For example, we know that every Borel set is Lebesgue measurable. But the continuous image of a Borel set need not be Borel. Is it Lebesgue measurable? It turns out that yes, but if we close the Borel sets under complements and continuous functions, will the resulting sets be Lebesgue measurable? Will they satisfy some sort of "continuum hypothesis"? Will they have the Baire property? And other questions, which are all quite natural, originated all sort of strange set theoretical objects and axioms which assert their existence.
And if you ask me, that is why we should learn set theory, and what its importance is. It allows us to better understand infinite objects, and the assumptions needed to better control their behavior.
