Kevin Ventullo's suggestion of Silverman's book is a very good one. The first examples
of Galois representations in nature are Tate modules of elliptic curves, and if you haven't
read about them in Silverman's book, you should.
If you have read Silverman's book, a nice paper to read is Serre and Tate's "On the good reduction of abelian varieties". It is a research paper, not a text-book, and is at a higher level than Silverman (especially in its use of algebraic geometry), but it has the merit of being short and beautifully written, and uses Galois representation techniques throughout.
One fantastic paper is Swinnerton-Dyer's article in Lecture Notes 350. Here he
explains various things about the Galois representations attached to modular forms.
The existence of the Galois representations is taken as a black box, but he explains
the Galois theoretic significance of various congruences on the coefficients of the
modular forms. Reading it is a good way to get a concrete feeling of what Galois
representations are and how you can think about and argue with them.
Another source is Ken Ribet's article "Galois representations attached to modular forms
with nebentypus" (or something like that) in one of the later Antwerp volumes. It presupposes some understanding of modular forms, but this would be wise to obtain anyway if you want to learn about elliptic curves, and again demonstrates lots of Galois representation techniques. It would be a good sequel to Swinnerton-Dyer's article.
Yet another good article to read is Ribet's "Converse to Herbrand's criterion" article, which is a real classic. It is reasonably accessible if you know class field theory, know a little bit about Jacobians (or are willing to take some results on faith, using your knowledge of elliptic curves as an intuitive guide), and something about modular forms. Mazur recently wrote a very nice article surveying Ribet's, available here on his web-site.
One problem with reading Serre is that he uses $p$-adic Hodge theory in a strong way,
but his language is a bit old-fashioned and out-dated (he was writing at a time when this
theory was in its infancy); what he calls "locally algebraic" representations would
now be called Hodge--Tate representations. To learn the modern formulation of and perspective on $p$-adic Hodge theory you can look at Laurent Berger's various exposes, available on his web-site. (This will tell you much more than you need to know for
Serre's book, though.)
For a two page introduction to Galois representation theory,
you could read Mark Kisin's What is ... a Galois representation? for a two-page
introduction.
Yet another source is the Fermat's Last Theorem book (Cornell--Silverman--Stevens), which
has many articles related to Galois representations, some more accessible than others.
The article of Taylor that Chandan mentioned in a comment is also very nice, although it
moves at a fairly rapid clip if you haven't seen any of it before.
Serre's article in Duke 54, in which he explains his conjecture about the modularity
of 2-dimensional mod p Galois representations, is also very beautiful, and involves
various concrete computations which could be helpful
One last remark: if you do want to understand Galois representations, you will need to have
a good understanding of the structure of the Galois groups of local fields (as described e.g. in Serre's book "Local fields"), in particular the role of the Frobenius element,
of the inertia subgroup, and of the significance of tame and wild inertia.
Best Answer
This is simple class field theory plus Galois theory. Consider a quadratic number field $K$ with class number divisible by $3$. For constructing an unramified cyclic cubic extension $L/K$, adjoin the cube root of unity, and denote the resulting field by $K'$. The Kummer generator of the Kummer extension $L' = K'(\sqrt[3]{\mu})$ must be an ideal cube for the extension to be unramified: $(\mu) = {\mathfrak m}^3$. Since $L'/K$ is abelian, Galois theory shows that the ideal class of ${\mathfrak m}$ must come from the quadratic subfield $F$ different from $K$ and ${\mathbb Q}' = {\mathbb Q}(\sqrt{-3})$. Thus the unramified cubic extensions of $K$ correspond roughly to the $3$-class group of $F$; any differences come from the fact that $\mu$ might be a unit.
The reflection theorem was found independently by Reichardt and then generalized by Leopoldt. For a dvi file of Scholz's article, see here.
Edit.Here's an English translation.