At the moment I'm reading the book Introduction to homotopy theory by Paul Selick. It is quite short but covers topics like spectral sequences, Hopf algebras and spectra. This is the first place I've found explanations (that I understand) of things like Mayer-Vietoris sequences of homotopy groups, homotopy pushout and pullback squares etc.. The author writes in the preface that the book is inteded to bridge the gap which the OP talks about.
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.
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