From the book of John M. Lee (2nd ed. 2013):

Chap.1 : Smooth mMnifolds

Chap.2 : Smooth Maps. In the section "Partition of Unity" the proofs of the theorems are less important.

Chap.3 : Tangent Vectors. The subsection "Categories and Functors" can be skipped.

Chap.4 : Submersions, Immersions, and Embeddings. The subsection "Smooth Covering Maps" is not included.

Chap.5 : Submanifolds. Until Prop. 5.35 p. 116.

Chap.6 : Sard's Theorem. Until p. 136 (The Whitney Approximation Theorem.). Understand Sard's theorem and Whitney's embedding theorem. Proofs can be skipped.

Chap. 8 : Vector Fields. Pages 174-189 (until "The Lie algebra of a Lie group"), and also page 190 about the general definition of a Lie algebra. See also Example 8.36, except (b) and the reference to a Lie group in (f).

Chap. 9 : Integral Curves and Flows. Pages 205-217 (until Flowouts)

Chap. 10 : Vector Bundles. Pages 249-257 (until Lemma 10.12)

Chap. 11 : The Cotangent Bundle. Pages 272-294.

Chap. 13 : Riemannian Metric. Pages 327-335 (until Prop. 13.19)