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.)
My suggestion, if you have really worked through most of Hartshorne, is to begin reading papers, referring to other books as you need them.
One place to start is Mazur's "Eisenstein Ideal" paper. The suggestion of Cornell--Silverman is also good. (This gives essentially the complete proof, due to Faltings, of the Tate conjecture for abelian varieties over number fields, and of the Mordell conjecture.) You might also want to look at Tate's original paper on the Tate conjecture for abelian varieties over finite fields,
which is a masterpiece.
Another possibility is to learn etale cohomology (which you will have to learn in some form or other if you want to do research in arithemtic geometry). For this, my suggestion is to try to work through Deligne's first Weil conjectures paper (in which he proves the Riemann hypothesis), referring to textbooks on etale cohomology as you need them.
Best Answer
I find Hodge theory pretty scary stuff with its compact inclusions of Sobolev spaces, pseudodifferential operators and parametrixes for elliptic differential operators. However it is very easy to use the results of Hodge theory as emanating from a black box. I remember how exhilarated I was by the argument that a Hopf surface, homeomorphic to $S^1 \times S^3$, could not be Kähler, and much less projective, just because its first Betti number is $b_1=1$. Whereas by Hodge theory a compact Kähler manifold $X$ has betti numbers $b_q(X)$ which are even whenever $q$ is odd.