Reference request: Best way of studying Loring Tu's "An Introduction to Manifolds" incompletely, but with restrictions

I've read this book and it is a very beautiful one. If you understand the derivative as a linear map already then you can skip the section on Euclidean Spaces and Foliations. In the Euclidean spaces section he does motivate very well where the notion of forms come from. He shows the construction for general finite dimensional vector spaces and dual vector spaces. The correspondence will be that the vector you are interested in is the tangent space and your differential forms will be coming from the co-tangent space. Also, if you are well equipped with the concept of orientation, you can skip a few sections there, however I don't think most people understand orientation on general manifolds.

I'm currently taking a course that covers those topics and I have been using that book. My recommendation would be to do chapters 5, 6, 8, 9, 11, 12, 14, 16 in that order. For chapter 12, I would skip the last three sections. That will take care of most of the stuff you are supposed to know. For the differential forms you can probably get away with reading Chapter 4 (differential forms on $\mathbb{R}^n$.) I just started learning about the De Rham Cohomology, so I don't have a very well-informed opinion yet, but there isn't very much to the definition. So if you only need to know the definition, then I wouldn't worry about it.