Michael Spivak: A comprehensive Introduction to Differential Geometry. Volume 1, Chapters 1-5, 7-8, 11 up to (but not including) the Thom class. (Although the Frobenius integrability theorem from Chapter 6 was proved in the class, the discussion was very brief and will not be asked at the exam.)
Additional recommended literature:
1. Bjørn Jahren's notes on De Rham cohomology (can be used as supplement to Chapter 11 in Spivak).
2. John M. Lee. Introduction to smooth manifolds.
3. D. Barden, C. Thomas. An introduction to differential manifolds.
4. I. Madsen, J. Tornehave. From calculus to cohomology.
5. R. Bott, L.W. Tu. Differential forms in algebraic topology.