From Loring W. Tu, "An Introduction to Manifolds":
§1-9, 11.1-11.4, 12.1, 12.2, §13-16, 17.1, 17.2, 17.5, 17.6, 18.1, 18.2, 18.4, 18.5-18.7, §19, 20.4, 21.1-21.4, §22, 23.3-23.6, §24, 25.1-25.3, 26.1, 26.2, 27.1, 27.4, §29, Appendices B.3, §C.
From Loring W. Tu, "Differential Geometry": Sections 15.4 - 15.7 (six pages). I recommend downloading this book from the library website and printing the few pages we need.
Supplementary reading: John M. Lee, "Introduction to Smooth Manifolds".