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.
Best Answer
Read about both simultaneously in the same book: Differential forms in Algebraic Topology !
(Bott was one of the best twentieth century geometers, and it shows in this extraordinary book)