Prerequisites for beginers
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. (Russian translation: Moscow, "Mir", 1972.)
Chapter 1: Rings and Ideals;
Chapter 2: Modules.
It is strongly suggested to solve exercises to these two chapters.

Maxim Kazarian, - Calculus on Manifolds - Moscow Independent University, MIM programme, Fall 2003.

John M. Lee, - Introduction to Smooth Manifolds, - Springer-Verlag, Graduate Texts in Mathematics, Vol. 218, 2003.
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. For this book in Russian see here.
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.