Cassels and Fröhlich is still the best reference for the basics of Class Field Theory, in my view. Cox's book, recommended by lhf, is also a good place to get motivation, historical and cultural background, and an overview of the theory.
Also the article What is a reciprocity law by Wyman is a helpful guide.
The key point to grasp is that there are two a priori quite distinct notions:
class fields, which are Galois extensions of number fields characterized by the fact that primes in the ground field split in the extension provided they admit generators satisfying certain congruence conditions (e.g. the extension $\mathbb Q(\zeta_n)$ of $\mathbb Q$, in which a prime $p$ splits completely if and only if it is $\equiv 1 \bmod n$); and abelian extensions, i.e. Galois extensions of number fields with abelian Galois group (e.g. the extension $\mathbb Q(\zeta_n)$ of $\mathbb Q$, whose Galois group over $\mathbb Q$ is isomorphic to $(\mathbb Z/n)^{\times}$).
The main result of class field theory is that these two classes of extensions coincide (as the example of $\mathbb Q(\zeta_n)$ over $\mathbb Q$ illustrates).
This fundamental fact can get a bit lost in the discussion of the Artin map, idèles, Galois cohomology, and so on, and so it is good to keep it in mind from the beginning, and to consider all the material that you learn in the light of this fact.
As for a more general road-map, that is a bit much for one question, but you could look at this guide on MO to learning Galois representations.
In one place, the theory of fibrations, I feel tom Dieck's book is superior. Thus in the first edition of Spanier, in the proof of Lemma 11, of Chapter 2, Section 7, Spanier writes down an extended lifting function $\Lambda$, but he does not prove it is continuous. I did not manage to prove it was continuous, and in fact found the function was not well defined. I sent a correction of the definition to Spanier, and this appeared in the second edition, but I still do not know how to prove $\Lambda$ is continuous. Have I missed something?
On the other hand Spanier's ideas on the construction of covering spaces are still referred today by experts, with the notion of what they call the Spanier group.
I find Section 3 of Chapter 7 of Spanier on "Change of base point" rather hard work, and I feel it can all be much easier done, and more generally, by using fibrations of groupoids. But tom Dieck's book does not use this method either.
Spanier gives van Kampen's theorem for the fundamental group of a simplicial complex as an exercise, while tom Dieck's book does give the statement of the theorem for a union of two spaces. Hatcher gives a more general theorem, for a union of many spaces, but none of these mention the fundamental groupoid on a set of base points.
I tend to agree with 313's answer that readers should look around, and find what is easier for them, in different aspects.
My copy of Spanier, dated 1966, had a price of \$15, but when I checked for inflation that was equivalent to \$111 today.
March 13: I add that neither book develops the algebraic theory of groupoids. For a relevant discussion on this, see https://mathoverflow.net/questions/40945/compelling-evidence-that-two-basepoints-are-better-than-one/46808#46808
Best Answer
If you would like to learn algebraic topology very well, then I think that you will need to learn some point-set topology. I would recommend you to read chapters 2-3 of Topology: A First Course by James Munkres for the elements of point-set topology. If you would like to learn algebraic topology as soon as possible, then you should perhaps read this text selectively. In particular, I would recommend you to focus mainly on the following (fundamental) notions, reading more if time permits:
I think that as far as algebraic topology is concerned, there are two options that I would recommend: Elements of Algebraic Topology by James Munkres or chapter 8 onwards of Topology: A First Course by James Munkres. The latter reference is very good if you wish to learn more about the fundamental group. However, the former reference is nearly 450 pages in length and provides a fairly detailed account of homology and cohomology. I really enjoyed reading Elements of Algebraic Topology by James Munkres and would highly recommend it. In particular, I think a good plan would be:
You will not need to know anything about manifolds to read Elements of Algebraic Topology but I believe that it is good to at least concurrently learn about them as you learn algebraic topology; the two subjects complement each other very well. I think a very good textbook for the theory of differentiable manifolds is An Introduction to Differentiable Manifolds and Riemannian Geometry by William Boothby (but this is a matter of personal taste; there are (obviously) many other excellent textbooks on this subject). The advantage of this textbook from the point of view of this question is that there is a flavor of algebraic topology present in one of the chapters.
I hope this helps!