I ran a minicourse on characteristic classes in summer 2020, specifically July 6–10. This is part of the more general summer minicourses program this summer; I encourage you to check out that website for more general information. I posted the course materials here: lecture notes, lecture slides, and, after each lecture, videos.
Monday: four perspectives on characteristic classes. Slides, exercises, video
Wednesday: Steenrod operations and Stiefel-Whitney classes. Slides, exercises, video
Thursday: Chern, Pontrjagin, and Euler classes; the splitting principle. Slides, exercises, video
Friday: Chern-Weil theory. Slides, video; no problem session today.
You can see the notes from a previous iteration here, but I made some changes. Most significantly, the fifth lecture was on Chern-Weil theory rather than characteristic classes in generalized cohomology theories. You should view the notes as a draft, and the slides as a reference; in particular, the exercises in the notes are not all good (some are boring, some are too hard, a few are wrong). Sorry about that.