Summer Minicourses 

Day 1: Orientation and Introduction. Fiber Bundles and $G$-bundles.

Day 1: Morphisms of $G$-bundles.

Day 2: Pullback Theorem, Fiber Bundle Construction Theorem and Associated Principal Bundle Functor.

Day 1: The Associated Bundle Construction and the equivalence between $\mathsf{Prin}_{G}$ and $\mathsf{Bun}_{G}^{F}$. Part 1 of smooth homotopy invariance

Day 2: Part 2 of smooth homotopy invariance. Topological homotopy invariance. Classifying spaces and the universal bundle.

Day 1: The Serre Spectral Sequence. First Computations and characteristic classes.

Day 2: More computations and the Splitting Principle.

This is the classic reference. It has a bit of everything. Be aware that the authors use some antiquated notation such as $\tau_{M}$ for $TM$.

Switzer's book is a classic. While it is a little old-fashioned in its description of spectra and the stable homotopy category, both from the $\infty$-categorical perspective and the modern symmetric monoidal point-set definitions, it's still a useful reference. The part relevant to this minicourse is the chapter on characteristic classics and Thom classes. Switzer develops this for a general (complex oriented) cohomology theory $E$.

Chapter IV discusses characteristic classes using de Rham cohomology. Besides this, the book has a bunch of other goodies.

This is another classic reference. It has a comprehensive treatment of connections in principal $G$-bundles and the curvature of such connections.

This is an interesting book. Tu cam sometimes be imprecise with definitions but when this happens it can be cross-checked. It should be seen as an alternative to Kobayashi and Nomizu's volumes.

From chapter 7 onwards, Spivak treats all the various ways of defining connections and shows they are all equivalent.

This is a great book and includes a lot more than just the basic differential topology we will need. One should be wary of some of the more delicate arguments Hirsch gives, especially wherever Hirsch appeals to his so-called globalization theorem is used. An incomplete list of errata may be found here but I'm also happy to talk about anything you suspect is a misprint or simply wrong. Read carefully.

This is another book that contains all of the differential topology we will need. Besides that, this book includes a lot of great stuff and this is its strength. Just as with Hirsch, one should also be wary of incorrect arguments and typos. Many arguments in this book are simply harder than they need to be or present things in a highly non-standard way. Read carefully.

The book doesn't have a detailed table of contents but Landweber and Ravenel have provided one here. My impression is that the book is really only for those serious about pursuing bordism but the appendix has a nice and careful treatment of some often neglected or incorrectly stated results in differential topology.

I can't confess to know much about this one. Genera are supposed to be important in theoretical and mathematical physics. Since genera are ring-homomorphisms from cobordism rings, it is unsurprising characteristic classes get involved.