I’m looking for a reference (preferably lecture notes or a book) that introduces De Rham’s cohomology and gives a proof of De Rham’s theorem. It would be nice if they included some applications too. I have some background in differential geometry and calculus on manifolds, and a bit of background in algebraic topology (simplicial homology mostly) and very basic category theory.
I’ve found the book ‘From calculus to cohomology: De Rham cohomology and characteristic classes’ by Ib Madsen and Jørgen Tornehave which seems quite helpful but it doesn’t include a discussion of De Rham’s theorem/singular (co)homology and its relation to De Rham’s cohomology.
You can find a treatment of De Rham cohomology and De Rham's theorem in chapters 17 and 18 of Lee's Introduction to Smooth Manifolds respectively.
I took an undergraduate course on differential geometry not long ago where this book was the main reference. I believe the categorical prerequisites - if any - should be very light.
In any case, as Jyrki says you should probably wait for an expert's opinion.