You may need to consult background material from time to time. Here are a few basic texts that you might find helpful

Advanced Calculus and Manifolds

C. H. Edwards, Advanced Calculus of Several Variables.
An excellent introduction to multivariable calculus, done right. Now available cheaply in the Dover series in advanced mathematics.
M. D. Spivak, Calculus on Manifolds.
A concise introduction to the basics of coordinate-free calculus. Contains an introduction to manifolds, differential forms, and Stokes' Theorem.
John Milnor, Topology from the Differentiable Viewpoint.
A short book, intended for undergraduates, but does an amazing job of elegantly and concisely introducing manifolds, Sard's theorem, and its applications to topology. A true classic.
Frank Warner, Foundations of Differentiable Manifolds and Lie Groups.
The first chapter is a good source for the basic definitions of the subject. The second chapter introduces tensors and differential forms. The third chapter introduces Lie groups. The fourth chapter treats integration on manifolds. We won't need the material in the fifth and sixth chapters.

Differential Geometry

M. D. Spivak, A Comprehensive Introduction to Differential Geometry, Vols. I–V.
Not entirely comprehensive, of course, but Volume 1 does a good job of exposing basic manifold theory, and the later volumes treat Riemannian geometry, the calculus of variations, geometric PDE, and characteristic classes.
Manfredo do Carmo, Riemannian Geometry
A more focussed introduction to the subject.
John Milnor, Morse Theory.
Besides being a wonderful introduction to the subject, this is a masterpiece of mathematical exposition. Deserves a place on the bookshelf of every geometer and every topologist.
R. Bott and L. Tu, Differential Forms in Algebraic Topology.
Useful for gaining basic expertise in calculation with differential forms and their applications in computations in differential topology.
Sigurdur Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces.
Everything you want to know about the title subjects.

Bryant, Chern, Gardner, Goldschidt, and Griffiths, Exterior Differential Systems.
The main source for proofs, etc. Unfortunately, currently out of print. (But stay tuned.)
Bryant, Chern, Gardner, Goldschidt, and Griffiths, Chapters 1-3 and 6-7 of Exterior Differential Systems.
This contains the basic material on exterior algebra and differential forms, the Frobenius, Cauchy, and Pfaff-Darboux Theorems, the Cartan-Kahler theorem (Chapter 3), Prolongation (Chapter 6), and Examples (Chapter 7)
Ivey and Landsberg, Cartan for Beginners.
A good introduction to the method of the moving frame and exterior differential systems.