Prerequisites
Beginners are supposed to be familiar with fundamentals of Commutative Algebra, Topology and Differential Geometry. For instance, all this can be found in the following sources:

M. F. Atiyah, I. G. MacDonald, - Introduction to Commutative Algebra, - Westview Press, 1969.
Chapter 1: Rings and Ideals;
Chapter 2: Modules.
It is strongly suggested to solve exercises to these two chapters.

John M. Lee, - Introduction to Smooth Manifolds, - Springer-Verlag, Graduate Texts in Mathematics, Vol. 218, 2003.
Appendix:
Topology (p. 540);
    • Continuity and convergency (pp. 541-543);
    • Hausdorff spaces (pp. 543-544);
    • Base and countability (pp. 544-545);
    • Subspace, product spaces, and disjoint unions (pp. 545-548);
    • Quotient spaces and quotient maps (pp. 548-549);
    • Open and closed maps (pp. 550-550);
    • Connectedness (pp. 550-552);
    • Compactness (pp. 552-553).
Chapter 1:
    •Topological Manifolds;
    •Smooth Structures;
    •Examples of Smooth Manifolds.
     (Chapter 1 is also available on the author's web page.)
Chapter 2:
    •Smooth functions and smooth maps (pp. 31-37);
    •Partitions of unity (pp. 49-57).
Chapter 3:
    •Tangent vectors (pp. 61-65);
    •Pushforwards (pp. 65-69);
    •Computations in coordinates (pp. 69-75);
    •Tangent vectors to a curves (pp. 75-77).
Chapter 4:
    •The tangent bundle (pp. 81-82);
    •Vector fields on manifold (pp. 82-89).
Chapter 6:
    •Covectors (pp. 125-127);
    •Tangent covectors on manifold (pp. 127-129);
    •The cotangent bundle (pp. 129-132);
    •The differential of a function (pp. 132-136);
    •Pullbacks (pp. 136-138).

Jet Nestruev, - Smooth manifolds and Observables . - Springer-Verlag, Graduate Texts in Mathematics, Vol. 220, 2002.
First chapters of this book will introduce you to the spirit of the school. People who have read this book and solved 70% of the exercises will be able follow the veteran courses.