I learned class field theory from the Harvard two-semester algebraic number theory sequence that Davidac897 alluded to, so I can really only speak for the "local first" approach (I don't even know what a good book to follow for doing the other approach would be, although I found this interesting book review which seems relevant to the topic at hand.).
This is a tough question to answer, partly because local-first/global-first is not the only pedagogical decision that needs to be made when teaching/learning class field theory, but more importantly because the answer depends upon what you want to get out of the experience of learning class field theory (of course, it also depends upon what you already know). Class field theory is a large subject and it is quite easy to lose the forest for the trees (not that this is necessarily a bad thing; the trees are quite interesting in their own right). Here are a number of different things one might want to get out of a course in class field theory, in no particular order (note that this list is probably a bit biased based on my own experience).
(a) a working knowledge of the important results of (global) class field theory and ability to apply them to relevant situations. This is more or less independent of the items below, since one doesn't need to understand the proofs of the results in order to apply them. I second Pete Clark's recommendation of Cox's book /Primes of the form x^2 + ny^2/.
Now on to stuff involved in the proofs of class field theory:
(b) understanding of the structure and basic properties of local fields and adelic/idelic stuff (not class field theory itself, but material that might be taught in a course covering class field theory if it isn't assumed as a prerequisite).
(c) knowledge of the machinery and techniques of group cohomology/Galois cohomology, or of the algebraic techniques used in non-cohomology proofs of class field theory. Most of the "modern" local-first presentations of local class field theory use the language of Galois cohomology. (It's not necessary, though; one can do all the algebra involved without cohomology. The cohomology is helpful in organizing the information involved, but may seem like a bit much of a sledgehammer to people with less background in homological algebra.)
(d) understanding of local class field theory and the proofs of the results involved (usually via Galois cohomology of local fields) as done, e.g. in Serre's /Local Fields/.
(e) understanding of class formations, that is, the underlying algebraic/axiomatic structure that is common to local and global class field theory. (Read the Wikipedia page on "class formations" for a good overview.) In both cases the main results of class field theory follow more or less from the axioms of class formations; the main thing that makes the results of global class field theory harder to prove than the local version is that in the global case it is substantially harder to prove that the class formation axioms are in fact satisfied.
(f) understanding the proofs of the "hard parts" of global class field theory. Depending upon one's approach, these proofs may be analytic or algebraic (historically, the analytic proofs came first, which presumably means they were easier to find). If you go the analytic route, you also get:
(g) understanding of L-functions and their connection to class field theory (Chebotarev density and its proof may come in here). This is the point I know the least about, so I won't say anything more.
There are a couple more topics I can think of that, though not necessary to a course covering class field theory, might come up (and did in the courses I took):
(h) connections with the Brauer group (typically done via Galois cohomology).
(i) examples of explicit class field theory: in the local case this would be via Lubin-Tate formal groups, and in the global case with an imaginary quadratic base field this would be via the theory of elliptic curves with complex multiplication (j-invariants and elliptic functions; Cox's book mentioned above is a good reference for this).
Obviously, this is a lot, and no one is going to master all these in a first course; although in theory my two-semester sequence covered all this, I feel that the main things I got out of it were (c), (d), (e), (h), and (i). (I already knew (b), I acquired (a) more from doing research related to class field theory before and after taking the course, and (f) and (g) I never really learned that well). A more historically-oriented course of the type you mention would probably cover (a), (f), and (g) better, while bypassing (b-e).
Which of these one prefers depends a lot on what sort of mathematics one is interested in. If one's main goal is to be able to use class field theory as in (a), one can just read Cox's book or a similar treatment and skip the local class field theory. Algebraically inclined people will find the cohomology in items (c) and (d) worth learning for its own sake, and they will find it simpler to deal with the local case first. Likewise, people who prefer analytic number theory or the study of L-functions in general will probably prefer the insights they get from going via (g).
I'm not sure I'm reaching a conclusion here: I guess what I mean to say is -- I took the "modern" local-first, Galois cohomology route (where by "modern" we actually mean "developed by Artin and Tate in the 50's") and, being definitely the algebraic type, I enjoyed what I learned, but still felt like I didn't have a good grip on the big picture. (Note: I learned the material out of Cassels and Frohlich mostly, but if I had to choose a book for someone interested in taking the local-first route I'd probably suggest Neukirch's /Algebraic Number Theory/ instead.) Other approaches may give a better view of the big picture, but it can be hard to keep an eye on the big picture when going through the gory details of proving everything.
(PS, directed at the poster, whom I know personally: David, if you're interested in advice geared towards your specific situation, you should of course feel welcome to contact me directly about it.)
I am tempted to stump for the centrality of Galois theory in modern mathematics, but I feel that this subject is too close to my own research interests (e.g., I have worked on the Inverse Galois Problem) for me to do so in a truly sober manner. So I will just make a few brief (edit: nope, guess not!) remarks:
1) Certainly when I teach graduate level classes in number theory, arithmetic geometry or algebraic geometry, I do in practice expect my students to have seen Galois theory before. I try to cultivate an attitude of "Of course you're not going to know / remember all possible background material, and I am more than willing to field background questions and point to literature [including my own notes, if possible] which contains this material." In fact, I use a lot of background knowledge of field theory -- some of it that I know full well is not taught in most standard courses, some of it that I only thought about myself rather recently -- and judging from students' questions and solutions to problems, good old finite Galois theory is a relatively known subject, compared to say infinite Galois theory (e.g. the Krull topology) and things like inseparable field extensions, linear disjointness, transcendence bases....So I think it's worth remarking that Galois theory is more central, more applicable, and (fortunately) in practice better known than a lot of topics in pure field theory which are contained in a sufficiently thick standard graduate text.
2) In response to one of Harry Gindi's comments, and to paraphrase Siegbert Tarrasch: before graduate algebra, the gods have placed undergraduate algebra. A lot of people are talking about graduate algebra as a first introduction to things that I think should be first introduced in an undergraduate course. I took a year-long sequence in undergraduate algebra at the University of Chicago that certainly included a unit on Galois theory. This was the "honors" section, but I would guess that the non-honors section included some material on Galois theory as well. Moreover -- and here's where the "but you became a Galois theorist!" objection may hold some water -- there were plenty of things that were a tougher sell and more confusing to me as a 19 year old beginning algebra student than Galois theory: I found all the talk about modules to be somewhat abstruse and (oh, the callowness of youth) even somewhat boring.
3) I think that someone in any branch of pure mathematics for whom the phrase "Galois correspondence" means nothing is really missing out on something important. The Galois correspondence between subextensions and subgroups of a Galois extension is the most classical case and should be seen first, but a topologist / geometer needs to have a feel for the Galois correspondence between subgroups of the fundamental group and covering spaces, the algebraic geometer needs the Galois correspondence between Zariski-closed subsets and radical ideals, the model theorist needs the Galois correspondence between theories and classes of models, and so forth. This is a basic, recurrent piece of mathematical structure. Not doing all the gory detail of Galois theory is a reasonable option -- I agree that many people do not need to know the proofs, which are necessarily somewhat intricate -- but skipping it entirely feels like a big loss.
Best Answer
Here are a few examples. In some, number theory provided an essential motivation. In the others, it plays a more direct role.
1) Are there nonisometric Riemannian manifolds that are isospectral (eigenvalues of the Laplacian match, including multiplicities)? An example was given by Milnor in the 1960s, which depended on prior work of Witt involving theta-functions (modular forms) of lattices. In the 1980s, Sunada created examples systematically by exploiting the analogy with the number theorist's construction of pairs of nonisomorphic number fields that have the same zeta-function. These number field pairs are found with Galois theory (find a finite group $G$ admitting a pair of nonconjugate subgroups having appropriate properties and then find a Galois extension of the rationals with Galois group isomorphic to $G$). There is a well-known analogy between Galois theory and covering spaces, and Sunada used this to translate the group-theoretic conditions for Galois groups into the setting of Riemannian manifolds. For more on this story, see the Wikipedia page here, where you'll see that the nonisometric isospectral pairs found between the work of Milnor and Sunada were closely related to other parts of number theory (quaternion algebras over the rationals).
2) Lens spaces are distinguished from each other using quadratic residues.
3) Knot theory uses continued fractions. See one of the answers to the MO question here. (Some of the other answers to that question could also be regarded as more applications of number theory, to the extent that you consider finite continued fractions to be number theory.)
4) The construction of Ramanujan graphs uses number theory. Also look here.
5) Frobenius proved that the only ${\mathbf R}$-central division algebras that are finite-dimensional are ${\mathbf R}$ and the quaternions. If you want to see infinitely many other examples of noncommutative division rings that are finite-dimensional over their centers, especially if you want examples that are more than 4-dimensional, you probably should learn number theory since the simplest examples come from cyclic Galois extensions of the rationals. Verifying the examples really work requires knowing a rational number is not a norm from a particular number field, and that amounts to showing a certain Diophantine equation has no rational solutions.
6) The classical induction theorems of Artin and Brauer about representations of finite groups were motivated by the desire to prove Artin's conjecture on Artin $L$-functions. Although number theory appears in the proof in the context of algebraic integers, the main point I want to make is that a conjecture from number theory provided an essential motivation to imagine the theorems might be true in the first place.
7) Several concepts of general importance in mathematics were originally developed within number theory. The most prominent example is ideals, which were first defined by Dedekind in his work on algebraic number theory. The first examples of finite abelian groups were unit groups mod $m$ and class groups of quadratic forms. The first finitely generated abelian groups to be studied as such were unit groups in number fields (Dirichlet's unit theorem). The first application of the pigeonhole principle was in Dirichlet's proof of the solvability of Pell's equation. The motivation for Steinitz's 1910 paper setting out a general theory of fields was Hensel's creation of $p$-adic numbers.