How do we study Iwasawa theory?
Iwasawa theory has its origins in the following counterintuitive insight of Iwasawa: instead of trying to describe the structure of any particular Galois module, it is often easier to describe every Galois module in an infinite tower of fields at once.
The specific example that Iwasawa studied was the $p$-Sylow subgroup of the class group of $K_n = \mathbb{Q}(\zeta_{p^n})$. It's naturally a $\mathbb{Z}_p$-module as well as a $G_n$ = Gal$(K_n/K_1)$-module, but the group ring $\mathbb{Z}_p[G_n]$ isn't very nice; it's not a domain, for instance. If we instead look at the inverse limits of the $p$ parts of the class groups of all the fields $K_n$ at once, as modules over $\mathbb{Z}_p[G_n]$, we get a module over the inverse limit $\varprojlim\mathbb{Z}_p[G_n]$. This ring is much easier to understand; it's a complete 2-dimensional regular local ring that is (non-canonically) isomorphic to a power series ring, and there is a strong structure theorem for modules over this ring. Using this structure theorem, Iwasawa proved many theorems about the class numbers of cyclotomic fields. For a simple example: $p$ divides the class number of one of the fields $K_n$ if and only if it divides the class number of all of the fields $K_n$.
There's an even bigger payoff to the theory: a profound connection with special values of $L$-functions. In the function field case, Weil had interpreted the Hasse-Weil $L$-function as computing the characteristic polynomial of Frobenius acting on the Jacobian of a curve. Iwasawa's idea was that the analogue for number fields should be the "characteristic ideal" of the ring $\varprojlim\mathbb{Z}_p[G_n]$ acting on ideal class groups. It turns out this characteristic ideal has a generator that is essentially the same as a $p$-adic $L$-function closely related to the ordinary Dirichlet $L$-functions. This was Iwasawa's "main conjecture" and is now a theorem. It implies the Herbrand-Ribet theorem and essentially every classical result relating cyclotomic fields and zeta values.
There have been many generalizations since but it's safe to call an area "Iwasawa theory" if it studies some Galois representation ranging over an infinite tower of fields and connects it to $p$-adic $L$-functions. The most fruitful Galois modules from the point of view of $L$-functions seem to be Bloch and Kato's generalized Selmer groups; the ideal class group can be interpreted as a Selmer group, and so can the classical Selmer group of an abelian variety. There's a lot of current research in this area.
To start reading, I recommend Washington's book on cyclotomic fields. Chapter 13 is fun and is a good use of some of the main techniques of Iwasawa theory. You don't need anything but the basic background in chapters 1-4 to read sections 1-4 of Chapter 13, which contain the types of theorem I was referring to in the first two paragraphs of this answer. The explicit computations in the first ten chapters also give the link to $p$-adic L-functions. If you know some algebraic number theory, you should be fine to read this book. I also strongly recommend Greenberg's PCMI notes on the Iwasawa theory of elliptic curves, which you can find here:
http://www.math.washington.edu/~greenber/Park.ps
If you're comfortable with class field theory, and have read the first few sections of Chapter 13 in Washington, then Coates and Sujatha's recent book, Cyclotomic Fields and Zeta Values, is a pleasure to read.
If I were you I'd read the following first: On $Z_l$-Extensions of Algebraic Number Fields, The Annals of Mathematics, Second Series, Vol. 98, No. 2 (Sep., 1973), pp. 246-326, By Iwasawa.
This THE paper every Iwasawa fan should read.
Then I'd take a look to On $p$-adic $L$-functions and cyclotomic fields Nagoya Math. J. Volume 56 (1975), 61-77, by Ralph Greenberg. Also check the continuation of Greenberg's paper.
The above papers are somehow the foundations of classical Iwasawa theory.
After the above, I highly recommend: Rational points of abelian varieties with values in towers of number fields. Invent. Math. 18 (1972), 183–266, by Mazur.
Check out Otmar Venjakob thesis, and his papers, to have an idea of non commutative Iwasawa theory.
About books I'd definitely have in my library Cyclotomic Fields and Zeta Values. Also Ralph Greenberg has been working on a book on Iwasawa theory, and I think you can find some chapters of it in his webpage.
Kato's lectures on the generalized Iwasawa conjecture adn how it fits into the general picture of zeta values, p-adic periods etc., are intended to be "joyfull lectures good for beginners" (unfortunately, the article contains only part I, does anyone know about part II?). An other overview is given in Kato's ICM lecture. Pottharst wrote a short note "What is Iwasawa theory about (in my opinion)?".