Let me attempt to answer this question. I should mention that I am not a research algebraic topologist. In fact, I am a student of algebraic topology and I hope to one day become a researcher in the area. I am currently on the path toward this goal.
Let me begin by saying that you are definitely on the right track by reading Hatcher's textbook. I think that the most fundamental topics of algebraic topology are covered in Hatcher's textbook and a knowledge of these topics will be very useful to you as a research mathematician no matter in which area of mathematics you specialize. I will assume that you have completed Hatcher's book and you are interested in further topics in algebraic topology.
I think the next step in algebraic topology (assuming that you have studied chapter 4 of Hatcher's book as well on homotopy theory) is to study vector bundles, K-theory, and characteristic classes. I think there are many excellent textbooks on this subject.
My favorite book in K-theory is "K-theory" by Michael Atiyah although some people object because they feel that the proof of Bott periodicity in this book is not very intuitive but rather long and involved (and I agree). However, you may as well assume Bott periodicity on faith if you read this book as the techniques used in proving Bott periodicity are not used or mentioned elsewhere in the book (although minor exceptions may show this statement to be false). I think a very slick proof of Bott periodicity is discussed in the paper "Bott Periodicity via Simplicial Spaces" by Bruno Harris. I would recommend you to read this paper if you are interested in a proof of Bott periodicity.
Alternatively, you may wish to learn from Hatcher's textbook entitled "Vector Bundles and K-theory" (available free online from his webpage) or the textbook by Max Karoubi entitled "K-theory: An Introduction". Hatcher's book discusses the image of the J-homomorphism (in stable homotopy theory) which is an important an interesting application of K-theory. I don't think that this is discussed in Atiyah's textbook. Similarly, Hatcher has a more detailed description of the Hopf-invariant one problem than that of Atiyah's book. Thus a good plan would be to read Atiyah's textbook and supplement it with a reading of the Hopf-invariant one problem and the J-homomorphism in Hatcher's book. Alternatively, you could read Karoubi's book which is much lengthier than the two (combined) but is an excellent textbook as well.
If you learn vector bundles and K-theory very well, then you should also learn the theory of characteristic classes. I believe that this is discussed in some detail in Hatcher's book (the same one entitled "Vector Bundles and K-theory") and the most basic properties of characteristic classes are proved. However, a more detailed discussion of characteristic classes can be found in the book entitled "Characteristic Classes" by Milnor and Stasheff. I would recommend reading the latter book if you have time and wish to learn about characteristic classes fairly thoroughly. Otherwise, the minimal treatment of characteristic classes in Hatcher's book is also sufficient in the short-term.
A good topic to learn about at this stage is spectral sequences. Spectral sequences furnish an extremely useful and efficient computational tool in algebraic topology. I can't really recommend the good book on spectral sequences because there are many but you might wish to look at "A User's Guide to Spectral Sequenes" by John McCleary and Hatcher's book on spectral sequences (available free online on his webpage).
Finally, you should now learn homotopy theory in more depth. An excellent place to do this is "Stable Homotopy and Generalized Homology" by Frank Adams. Unfortunately, this is as far as I can advise you because this is as far as I have progressed in algebraic topology. I think once you finish the book "Stable Homotopy and Generalized Homology" by Frank Adams the next step could be to start reading research papers (which you have to do sooner or later). Of course, advice on reading research mathematics papers is long and involved so I won't go into details in this answer as we are discussing algebraic topology. But, the books I suggested should keep you busy at least in the short term.
I hope this helps!
My personal opinion is that category theory is like set theory; it's a language, everyone should know the basics, and everything in the "basics" is essentially trivial. Here "basics" for set theory means subsets, products, power sets, and identities like $f^{-1}(\bigcap A) = \bigcap f^{-1}(A)$. For category theory, I think "basics" means:
- categories, functors, natural transformations;
- duality;
- basic constructions like product categories, comma categories (at least over- and under-categories), and functor categories;
- universal properties, representable functors, and the Yoneda lemma/embedding;
- limits and colimits;
- adjunctions.
Basically the first 4 chapters of Mac Lane (ignoring the stuff about graphs and foundations). One could probably add "abelian categories" to that list, but I think a homological algebra text is a better place to learn that.
Best Answer
The list of possible topics that you provide vary in their categorical demands from the relatively light (e.g. differential topology) to the rather heavy (e.g. spectra, model categories). So a better answer might be possible if you know more about the focus of the course.
My personal bias about category theory and topology, however, is that you should mostly just learn what you need along the way. The language of categories and homological algebra was largely invented by topologists and geometers who had a specific need in mind, and in my opinion it is most illuminating to learn an abstraction at the same time as the things to be abstracted. For example, the axioms which define a model category would probably look like complete nonsense if you try to just stare at them, but they seem natural and meaningful when you consider the model structure on the category of, say, simplicial sets in topology.
So if you're thinking about just buying a book on categories and spending a month reading it, I think your time could be better spent in other ways. It would be a little bit like buying a book on set theory before taking a course on real analysis - the language of sets is certainly important and relevant, but you can probably pick it up as you go. Many topology books are written with a similar attitude toward categories.
All that said, if you have a particular reason to worry about this (for instance if you're worried about the person teaching the course) or if you're the sort of person who enjoys pushing around diagrams for its own sake (some people do) then here are a few suggestions. Category theory often enters into topology as a way to organize all of the homological algebra involved, so it might not hurt to brush up on that. Perhaps you've already been exposed to the language of exact sequences and chain complexes; if not then that would be a good place to start (though it will be very dry without any motivation). Group cohomology is an important subject in its own right, and it might help you learn some more of the language in a reasonably familiar setting. Alternatively, you might pick a specific result or tool in category theory - like the adjoint functor theorem or the Yoneda lemma - and try to understand the proof and some applications.