Here are some comments on the last two chapters of Introduction to Topological Manifolds (ITM) and suggestions for reading certain sections of Introduction to Smooth Manifolds (ISM).

In ITM Chapter 12, you should read at least to the middle of p. 318. The rest of the chapter is optional, but at least browsing it is recommended. "Proper Actions" are presented again in ISM and we will make serious study of them, so you could look at the material in ITM Chapter 12 as a preview. Example 12.28, Lens spaces, and the Universal Covering Spaces for (Compact) Surfaces are objects you should at least have a passing acquaintance with. The latter involves a bit of complex analysis.

ITM Chapter 13 is completely optional. For a very brief view of some aspects of homology, now or later, read the introduction, pp. 339-340, and the section on homology of spheres, pp. 364-369 (skipping the proofs, of course). Glance through the chapter and read anything else that interests you.

• Preface: at least read the paragraph on notation on p. vii: "I should say ... efficiency later."
• Appendix A: You may use either this appendix or ITM when you need to cite topological results. There is one new topic, Lipschitz continuity (starting after Exer. A.46, through Exercise A.49) which you may consult when and if needed.
• Appendices B & C: Look these over, if any section doesn't seem immediately and completely obvious, read more carefully and do the exercises. Exceptions: You may skip Proposition B.57 and its proof, and may skip the proofs of the Inverse and Implicit Function Theorems. You may postpone the section on multiple integrals until we get to Chapter 16.
• Appendix D: Consult as needed in Chapter 9 and later.
• Chapter 1: There is little or nothing new for us in pp. 3-10, so you may skim this quickly. Starting at the bottom of p. 10, read as usual: Your first read may skip details, but be sure to work through all proofs and exercises sometime soon after that.
• Chapter 6: For many of the results in this Chapter, we will have occasion to cite the results, but the techniques used in the proofs will not be important for us. Here is a guide to what you should take away from this chapter.
• From the first section "Sets of Measure Zero," you should understand that the concept of a set of measure zero is invariant under diffeomorphism, so makes sense on manifolds.
• Know the statement of Sard's Theorem, Theorem 6.10. OK to skip proof.
• Whitney Embedding Theorem section: skip to p. 134, and read Theorem 6.15 and the first part of the proof, the case of a compact manifold. OK to skip rest of proof and variations of the theorem.
• Whitney Approximation Theorems: The one for functions, Theorem 6.21, and its proof are so basic that it sometimes appears on the prelim. (The expectation is that students will prove it from scratch, not quote the result.) We will need some of the other approximation theorems in Chapter 21, but you can postpone reading them until we need them.
• Read the subsection on Tubular Neighborhoods (pp. 137-141) and the section on Transversality (pp. 143-147). These concepts and proof techniques are more central to our course than the rest of the chapter.
• Chapter 9: The chapter is long; for the RReport, stop early if you get saturated. The proofs of Thms. 9.12 and 9.20, each 2 pages, are technical and probably are better skipped on first read. Instead, think about what the results mean. The following are optional: "Flows amd Flowouts on Manifolds with Boundary" (mid p. 222 to mid p. 227) and "Time-Dependent Vector Fields" and "First order PDEs" (p. 236 through end of chapter).
• Chapter 13: One section you should read because we will use it late in Chapter 14: "The Tangent-Cotangent Isomorphism," pp. 341-343. This section is independent of the rest of the chapter. Addressing this section in your RReport is optional.
The rest of the chapter is optional reading. Recommended before Stephen's guest lecture on Riemannian geometry on 4/22: the beginning of the chapter through Proposition 13.9 on p. 331, the discussion of isometry on the middle of p. 332, and the start of the subsection on Riemannian Submanifolds (at least the half page on p. 333, more if you are enjoying it or hope to teach vector calculus soon).
Comments on the rest of the chapter: The material on "flat" manifolds and metrics is a bit contrived, and is done with better tools in 547. The sections on the Riemannian distance function and Psuedo-Riemannian metrics are standard introductions to those topics.