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!
Chapter 1 of Hatcher corresponds to chapter 9 of Munkres. These topology video lectures (syllabus here) do chapters 2, 3 & 4 (topological space in terms of open sets, relating this to neighbourhoods, closed sets, limit points, interior, exterior, closure, boundary, denseness, base, subbase, constructions [subspace, product space, quotient space], continuity, connectedness, compactness, metric spaces, countability & separation) of Munkres before going on to do 9 straight away so you could take this as a guide to what you need to know from Munkres before doing Hatcher, however if you actually look at the subject you'll see chapter 4 of Munkres (questions of countability, separability, regularity & normality of spaces etc...) don't really appear in Hatcher apart from things on Hausdorff spaces which appear only as part of some exercises or in a few concepts tied up with manifolds (in other words, these concepts may be being implicitly assumed). Thus basing our judgement off of this we see that the first chapter of Naber is sufficient on these grounds... However you'd need the first 4 chapters of Lee's book to get this material in, & then skip to chapter 7 (with 5 & 6 of Lee relating to chapter 2 of Hatcher).
There's a crazy amount of abstract algebra involved in this subject (an introduction to which you'll find after lecture 25 in here) so I'd be equally worried about that if I didn't know much algebra.
These video lectures (syllabus here) follow Hatcher & I found the very little I've seen useful mainly for the motivation the guy gives. If you download the files & use a program like IrfanView to view the pictures as you watch the video on vlc player or whatever it's much more bearable since you can freeze the position of the screen on the board as you scroll through 200 + pictures.
I wouldn't recommend you treat point set topology as something one could just rush through, I did & suffered very badly for it...
Best Answer
Answers to this sort of question obey an uncertainty principle: The more precisely I outline a manageable "syllabus", the less likely it is to match your specific needs and interests. That said, here's a rough outline based on Hatcher's Algebraic Topology that can be adjusted as needed.
General advice: Most of Hatcher's sections begin with a paragraph or two regarding the usefulness of or intuition for the topic at hand. I strongly recommend revisiting these gems of wisdom as you move through the text; they offer a review of the central ideas and a sort of road map for the subject. Also, given the volume of new ideas, I emphasize the standard advice of skimming proofs during an initial read of each subsection and then going back to understand the details of the arguments. Finally (but first, really), read the preface! It's an instruction manual for how to read the book and understand its place in the broader literature.
Chapter 0: Some Underlying Geometric Notions. Read the whole thing at first, but don't expect yourself to memorize the details of every construction in one read. Since it sounds like you know the basics of homotopy of maps, homotopy equivalence, contractibility, et cetera, your goal here should be to make sure you're comfortable with CW complexes and their homotopy properties (e.g. homotopy equivalence under collapsing contractible subcomplexes, equivalence of homotopic attaching maps, CW pairs possessing the homotopy extension property).
Exercises: 1-6, 10, 17, 19
Chapter 1: The Fundamental Group. If you feel pretty good about fundamental groups, van Kampen's theorem, classification of covering spaces, just skim $\S$1.1-1.3 and review the statements of theorems/propositions/corollaries and definitions typeset in bold. Read the examples and "Applications to Cell Complexes" in $\S$1.2, though you can skim the more detailed discussion/examples of covering space actions on a first read. The appendices $\S$1.A-1.B are fun and have plenty of applications but are not strictly necessary. (But do read the definition of a $K(G,1)$ and statement of Proposition 1B.9.)
Exercises: $\S$1.1 -- 5, 6, 8, 10, 13, 16; $\S$1.2 -- 4, 6, 8, 9, 10, 15, (+22 if you like knots); $\S$1.3 -- All great, so do what time permits.
Chapter 2: Homology. Almost nothing to skip in $\S$2.1. Many people skim the details of barycentric subdivision/Proposition 2.21 in the section on the Excision Theorem, but you definitely need to understand the statement and the way it is used to prove the main theorem. Lots of courses skip many of the examples in $\S$2.2. You should read about cellular homology, but your goal should be computational intuition and not mastery of the details, such as why it is equivalent to the other theories. Learn Mayer-Vietoris very well, homology with coefficients well, and get a feel for Euler characteristic. Some people glaze over $\S$2.3, but it helps you organize your thoughts about homology and provides an introduction to category theory -- something you need sooner or later. For $\S$2.A, at least internalize the central statement: $H_1(X)$ is the abelianization of $\pi_1(X,x_0)$. Many courses skip $\S$2.B-2.C.
Exercises: $\S$2.1 -- 4, 5, 7, 11, 13, 15, 16, 20, 22, 27, 29, 30; $\S$2.2 -- 4, 12, 14, 15, 31, 32, 41; ($\S$2.3 -- Try all, if you read the section.)
Chapter 3: Cohomology. In a first reading of $\S$3.1, you can skip the details of the discussion on pages 191-195 beginning after "Our goal is to show that the cohomology groups..." and ending before "Summarizing, we have established...". For $\S$3.2, some people only learn the statement of the Künneth formula during a first reading. Skip all but the first theorem in "Spaces with Polynomial Cohomology" if you'd like. As for $\S$3.3, there's much debate about how much detail a first reading about Poincaré duality should include. I suggest reading all of the introduction and "Orientations and Homology", plus all of "The Duality Theorem" up to (and including) the statement of Poincaré duality for closed manifolds. Also learn the results (but not necessarily the proofs) in "Other Forms of Duality". For $\S$3.A, just learn the major results and computational propositions/corollaries. Same, but less so, for $\S$3.B. Skip $\S$3.C-3.E and $\S$3.G-3.H for now, but maybe check out $\S$3.F for limits.
Exercises: $\S$3.1 -- 5, 6, 8, 13; $\S$3.2 -- 1, 3, 7; $\S$3.3 -- 2, 3, 5, 7-10, 16, 24, 30, 31, 32, 33
If you have time: Learn the basics of homotopy theory. Try to read $\S$4.1 (which is full of really useful ideas), plus some of the other fundamentals like the Hurewicz theorem, fibrations and fiber bundles, and the connection between singular cohomology and Eilenberg–MacLane spaces.