If you wish to approach Category Theory from the viewpoint of a programmer, not a mathematician, I suggest Bartosz Milewski's book Category Theory for Programmers. For this, all you need is some previous programming experience. He uses C++ and Haskell iirc but as long as you can read snippets of code, you'll be fine.
I am suggesting this since you said you want to better understand functional programming. Category Theory, as mathematicians look at it, is an extremely abstract field. If you want to do pure math related stuff in Category Theory, and only then, I would say important prereqs are Abstract Algebra and Topology. I believe the motivation for Category theory lies in Algebraic Geometry and Algebraic Topology, but you definitely don't need to be an expert on these to learn it.
I am suggesting this since you said you want to better understand functional programming. Category Theory, as mathematicians look at it, is an extremely abstract field. If you want to do pure math related stuff in Category Theory, and only then, I would say important prereqs are Abstract Algebra and Topology. I believe the motivation for Category theory lies in Algebraic Geometry and Algebraic Topology, but you definitely don't need to be an expert on these to learn it.