Allow me to "add my two cents" to the detailed answer given by @ Adam Hughes. He rightly stresses that "if you ever wanted to do something at a full research level, certainly you should know as much as you can", and so you must get acquainted with both the two main approaches to CFT (I put apart Neukirch's, which I dare say is not useful to a beginner):
1) The approach via ideals, which was the first historically, starting with Kronecker-Weber (1886-87) and culminating with Takagi's main theorems (1920) on "K-modulii", "ray class fields", "conductors", ramification, decomposition ... , and Artin's reciprocity law (1927). In the excellent account by D. Garbannati, CFT summarized, Rocky Mountain J. of Math. 11, 2 (1981), this period is referred to as "classical global CFT". In this approach, local CFT is derived from global CFT. To get an idea of how to use this classical machinery, see e.g. G. Gras' book, CFT:From theory to practice, Springer (2005)
2) The subsequent developments via Chevalley's "idèles", referred to as "post - world war CFT" (op. cit.), in which global CFT is derived from local CFT. Not only Local CFT is easy (M. Hazewinkel, Adv. in Math. 18 (1975)) but the classical global CFT presents "aesthetically unpleasing aspects" (Garbanati, op. cit.), such as the "defining modulii" which vary with the finite abelian extensions of a given number field K and prevent to go smoothly to infinite extensions (such as in K-W's theorem).
Actually this change of perspective goes beyond mere technique, it ingrains in CFT (as more generally in arithmetic) the so called "local-global principle", which asserts, roughly speaking, that a certain property (not all !) over a number field K holds globally iff it holds locally over all its completions (p-adic as well as archimedean). This is where cohomology comes into play, because it appeared as the most convenient way to make the local-global machinery work. Essentially, the main new ingredient is the Brauer group of K. When expressed cohomologically, the local-global principle applies to it in a crystal clear manner.
One last word about cohomology, which nowadays completely pervades both algebraic and arithmetic geometry. Practically, you can consider it as a mere tool, at the same level as e.g. Taylor's formula in analysis, i.e. an approximation device. Taylor's expansion allows to approximate the value of an analytic function. The cohomological functor does the same for an algebraic object, by deriving from a short exact sequence an infinite exact sequence of cohomology groups