The whole set is expensive, so perhaps if you stick to your decision of acquiring it you should buy only volume one at first.
As for the proofs, his proofs and methods are not all short and slick. In fact, he is rarely short and slick. He prepares the chain rule for three sections before getting into the actual result, and by then it will feel very naturally. The book has plenty of computational examples. He makes frequent interludes to help the reader acquire the necessary background knowledge if he does not have it already. For instance, the chapter on differentiable mappings opens up with a whole section on linear algebra.
This is the chapter breakdown of volume one:
Chapter 1 deals with continuity, which is basically topology in $\mathbb{R}^n$. It is short by its nature, most proofs of the basic results are very simple. He still presents nice examples and computations.
Chapter 2 deals with differentiable mappings. He takes his time to develop linear algebra tools and every proposition so that the whole idea of differentiation becomes natural. He discusses higher-order derivatives and a thorough examination of the interchange of limit operations (limits and derivatives, limits and integration, integration and derivatives) with a very nice set of examples.
Chapter 3 deals with the inverse and implicit function theorems. It follows a nice format: one section for motivation, one section for the proof (which is fairly drawn out) and one section for applications. Thus one learns not only why the theorems are important, but the proofs are detailed and you learn to apply it in various conditions.
Chapter 4 deals with manifolds in $\mathbb{R}^n$. He defines manifolds in four ways and exemplifies the situations where each description is most useful. He follows the format of the third chapter when he discusses the immersion and submersion theorems. His discussion of Morse's lemma is okay.
Chapter 5 deals with tangent spaces. This chapter is basically an introduction to differential geometry. It opens up by discussing tangent spaces and giving NINE examples of construction of nontrivial tangent spaces (such as the cycloid and Descartes's folium). He proves and applies the method of Lagrange multipliers and revisits critical points. Then you deal with more differential geometry specific topics such as Gaussian curvature, curvature and torsion of curves, linear Lie groups and Lie algebras.
The problem sets for each chapter are varied and very complete. Most difficult problems are very detailed and broken down, as much as eleven items. He fits nice discussions and investigations into the exercise chapters, such as:
Hilbert's space-filling curve, Weierstrass's approximation theorem on $\mathbb{R}$, an analog of Rolle's theorem, Casimir and Euler operators, Cartan decomposition, Laplace integrals, Airy function, partial derivatives in arbitrary coordinates, confocal coordinates, moving frame, gradient in arbitrary coordinates, divergence in arbitrary coordinates, Laplacian in arbitrary coordinates, the cardioid, quadrics, the rotation group of $\mathbb{R}^n$, the special linear group, the Hopf fibration, stereographic projection, Steiner's Roman surface, Cayley's surface, Whitney's umbrella, the tangent bundle of a submanifold, the lemniscate, the astroid, Diocles's cissoid, conchoid and trisection of angles, Villarceau's circles, spherical trigonometry, Mercator projection of sphere onto cylinder, Steiner's hypocycloid, and much more.
That's my argument.
I browsed my library and I found another book which, for differential forms, might be better suited to you: Steven Weintraub Differential Forms, Theory and Practice. :)
