In addition to other good answers: I'd tend to recommend taking a cohomological approach to classfield theory as a second pass through the subject, so that you already know the down-to-the-metal number-theoretic facts, and can focus on re-interpreting them in (co-)homological terms. Or, oppositely, if one already is well acquainted with cohomological things, then it's "just" seeing how they may apply to classfield theory. "One thing at a time" is often best, in my own experience.

For practice, be sure you can rewrite Hilbert's theorem 90 homologically.

I'd also strongly recommend not taking the most traditional approach to group cohomology (with "(in)homogeneous bar resolutions" as the definition) ... as opposed to a more modern (1950s!?!) approach via derived functors of very natural functors, namely, the (co-)fixed-vector functors $M\to M^G$ and $M\to M_G$. This shows the commonality with sheaf cohomology, Lie-group (co)homology, Lie-algebra (co)homology, $(\mathfrak g,K)$-cohomology and lots of other similar things. Yes, for computations, sometimes we do want explicit (small) projective or injective resolutions, but it's not optimal to take those as the definition... especially since the whole general machine is well established.

In particular, some of the colorfully-named homological mechanisms in classfield theory are really special cases of far more general homological phenomena. Herbrand quotient. Dimension shifting. In particular, to imagine a necessity of seeing the number-theoretic content in these mostly homological ideas is misguided. At the same time, insufficient homological chops can leave a person "stranded" doing ugly approximations thereof, thinking that it's supposed to be "number theory". :)

In addition to other good answers: I'd tend to recommend taking a cohomological approach to classfield theory as a second pass through the subject, so that you already know the down-to-the-metal number-theoretic facts, and can focus on re-interpreting them in (co-)homological terms. Or, oppositely, if one already is well acquainted with cohomological things, then it's "just" seeing how they may apply to classfield theory. "One thing at a time" is often best, in my own experience.

For practice, be sure you can rewrite Hilbert's theorem 90 homologically. EDIT: per LSpice's apt comment, it would be most accurate to say "cohomologically" here, just so no one is inadvertently misled. :)

I'd also strongly recommend not taking the most traditional approach to group cohomology (with "(in)homogeneous bar resolutions" as the definition) ... as opposed to a more modern (1950s!?!) approach via derived functors of very natural functors, namely, the (co-)fixed-vector functors $M\to M^G$ and $M\to M_G$. This shows the commonality with sheaf cohomology, Lie-group (co)homology, Lie-algebra (co)homology, $(\mathfrak g,K)$-cohomology and lots of other similar things. Yes, for computations, sometimes we do want explicit (small) projective or injective resolutions, but it's not optimal to take those as the definition... especially since the whole general machine is well established.

In particular, some of the colorfully-named homological mechanisms in classfield theory are really special cases of far more general homological phenomena. Herbrand quotient. Dimension shifting. In particular, to imagine a necessity of seeing the number-theoretic content in these mostly homological ideas is misguided. At the same time, insufficient homological chops can leave a person "stranded" doing ugly approximations thereof, thinking that it's supposed to be "number theory". :)

One more EDIT: it has always amazed me that in local or global classfield theory, the "reciprocity law" map is a cup product in cohomology!!! With a sort of "fundamental class"... and so on. But I'm glad I already knew about quadratic reciprocity, Kronecker-Weber, and so on... :)