I have spent the last two or so years making myself well rather well acquainted with the foundational aspects of differential topology/geometry. I have also spent the last year taking a two course graduate sequence in abstract algebra, which was based on Aluffi's Chapter zero book. We ended up going over a good amount of category theory, and worked up to an introduction to homological algebra. Personally, I like the categorical presentation of things quite a bit, and would prefer a reference which uses category theory explicitly from the get go.
In the fall I will be taking a more comprehensive course in algebraic topology, but for the moment, I have a personal need to understand Stiefel-Whitney classes, Pontrjagen classes, and intersection forms. I feel like I have good understanding of de Rham cohomology, but once I start seeing things like $H^*(BSO(4),\mathbb{Q})$ in the context of four manifolds, I get very confused of what this cohomology group even is.
Essentially, I am looking for a reference, or multiple references, which I can use to get a working understanding of what these, and characteristic classes in general, are, without having to read some 200 odd pages of material before getting to this. My thinking is that in the fall I can shore up any gaps from taking this route. Please forgive my ignorance if this is simply not possible.
Any references/confirmation of impossibility would be greatly appreciated.
The canonical reference for characteristic classes is Milnor and Stasheff. An extremely readable classic, it starts with basic notions and properties of vector bundles and Grassmannians, then move on to various characteristic classes. Milnor and Stasheff builds characteristic classes using Steenrod squares (briefly explained in Hatcher chapter 4), but it is possible to take Steenrod squares as a black box. The problem with Milnor and Stasheff is that it doesn't deal with the smooth side of things sufficiently (my personal opinion). Another problem with Milnor and Stasheff is that it is not entirely self contained, for example you might want to learn obstruction theory from elsewhere (e.g. Steenrod's bundle book) if you want to understand parts of Milnor and Stasheff. Nevertheless this is the book I started with and probably what you are looking for.
Kobayashi and Nomizu is another well known reference for differential geometry, and it has a chapter (volume II, you can directly jump to this chapter if you have sufficient background) on characteristic classes. The tome is precisely written but quite demanding, probably read this after Milnor and Stasheff. Also one should definitely not read Kobayashi and Nomizu cover to cover because of its huge breadth, I personally only read chapters that are directly relevant to what I need (chapters on connections and characteristic classes).
Bott and Tu is a more elementary reference, probably a bit too simple for you but can be an enjoyable read. Bott and Tu also treats both the algebraic topology aspects and differential geometry aspects in a single book.