You can start studying Algebraic Topology without knowing anything about Category Theory. It is also true that Algebraic Topology consists in traveling (I mean transforming Topological problems in Algebraic ones, solving the second ones and then come back to the topology land and try to deduce some consequences of the solution for the algebraic problem). However, for example, you could start by working on chapters 0 and 1 of Hatcher without necessity of Category Theory.
Anyway some Category Theory is always useful. If you want a very brief overview designed for Algebraic Topology, then I recommend you to take a look at chapter 0 of Rotman's book: An Introduction to Algebraic Topology. He gives a very quick picture of the basis of Category Theory and then he develops it during the book when he needs it.
Once you are confident with some Category Theory Tammno Tom Dieck book is great. In fact, more Category Theory you know, more related the stuff is in your mind. But from my own experience, the Geometry is what matters and motivates what you do. The Category Theory and Algebra are the ways to make the statements trivial and provide you with a powerful machinery.
To sum up. Start by looking at the Geometry, once you have understood it, turn to Categories so all will follow easily and will be more related.
If you give more details about what you are studying, then I will be able to give you more accurate recommendation.
I hope this helps.