John M. Lee: Introduction to Smooth Manifolds, Second Edition, Springer.
Other recommended literature:
1. Bjørn Jahren's notes on de Rham cohomology.
2. M. Spivak: A Comprehensive Introduction to Differential Geometry, Volume 1.
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.