I would like to learn the language of Differential Geometry formally. I have a basic understanding of Non Abelian Gauge Theory from my study of physics. However I don't understand the language used by mathematicians(and physicists) properly. I am looking for books that start with basic notions In differential geometry, principle g bundles, connection, curvatures etc. After that I would like to learn about about sheaves, Cech cohomology etc.
I would like the presentation to be sufficiently abstract while maintaining its root in structures used in physics. I am not interested(At this moment) in the most abstract way to treat the subject.
I would eventually like have the knowledge to understand articles on N Lab(and references linked) and lectures such as Lectures on Higher structures In M theory to name a few.
Feel free to recommend books that would be useful for students of different levels I suspect that people with varying backgrounds would be interested in studying these topics. It would be nice to have a compilation of books that teach this aspect of mathematical physics.