Can Somebody Please Outline a Reading Course For Me in Algebraic Topology

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 §§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 §§1.2, though you can skim the more detailed discussion/examples of covering space actions on a first read. The appendices §§1.A-1.B are fun and have plenty of applications but are not strictly necessary. (But do read the definition of a