My background in topology is the first 6 chapters of Munkres's "topology" and in algebra Herstein's "Topics in Algebra". Both of them I self studied.
A look at the table of contents of Bredon's "Topology and Geometry" got me to really want to read it (I really like the emphasis on the differential side of things). I don't have any background in differential topology/geometry though...
What are the prerequisites I should be familiar with before tackling this book?
You should read Milnor's topology from a differentiable viewpoint (two or three times) first, then Bott/Tu. Then you are good to go.