How to read Hatcher's Algebraic Topology?

How good is your background in topology? For example, have you mastered Munkres' book?

The point of view in Hatcher's book requires you to have already mastered several important topics in topology including these two key topics:

• Quotient maps and quotient topologies, which are the key to CW complexes;
• Homotopies, which are the key to deformation retractions and homotopy equivalences.

Just as an example, I would expect someone who has mastered Munkres' book to be able to write down an explicit formula for a subset of $\mathbb R^2$ that is homeomorphic to one of the graphs in that discussion of Hatcher, to write down an explicit formula for a specific deformation retraction from a disc with two holes to that graph, and to write down the specific formulas for the homotopies needed to prove that map to be a deformation retraction. You can think of that discussion of Hatcher as a "prerequisite quiz" which tests whether you have learned what you need to learn about homotopies.

So, if you find yourself unable to write down such maps and such homotopies, or if you have any other deficiencies in those two topics, or in any other basic topics in topology, you should shore up those topics with another book such as Munkres as you proceed into Hatcher's book.

As others have suggested, one solution could be to try another book. There are many possibilities out there, but a good one for beginners is Lee's Introduction to Topological Manifolds. Even though it is about manifolds, it takes plenty of time to introduce key topics from general and algebraic topology. If your background in general topology is sufficiently strong, you can go straight to Chapter 5 on cell complexes (in the second edition, the focus is on CW complexes) and work on from there. Chapter 10 is on the Seifert-Van Kampen theorem. Lee is very careful and thorough in his presentation, which could help you with the gaps you have encountered.