I've quickly scanned through the book, and here's what I could discern. I welcome edits from other people who are more familiar with the Goldstein, Poole, & Safko than I am.

As you correctly surmise, Chapters 1 & 2 should be learned before anything else. Beyond this, the following chapters & sections rely on previous chapters other than #1 and #2:

• Chapter 5 (Rigid-Body Equations of Motion) relies extensively on Chapter 4 (Kinematics of Rigid-Body Motion). Section 5.8 (Precession of Equinoxes) would also benefit from familiarity with Chapter 3 (The Central Force Problem).
• Section 7.7 (Relativistic Collisions) would benefit from familiarity with Sections 3.10–11 (Scattering Problems).
• Section 8.4 (Hamiltonian Formulation of Relativistic Mechanics) requires Chapter 7 (Special Relativity).
• Chapter 9 (Canonical Transformations) requires Chapter 8 (The Hamilton Equations of Motion). Section 9.7 (Angular Momentum Poisson Brackets) requires Chapter 4.
• Chapter 10 (Hamilton-Jacobi Theory & Action-Angle Variables) requires Chapters 8 & 9. Section 10.5 & 10.8 (which apply these tools to the Kepler Problem) requires Sections 3.7–8 (the Kepler Problem).
• Chapter 11 (Classical Chaos) draws on Chapters 3 & 10 for examples, particularly in its presentation of the KAM theorem (Sections 11.1–2).
• Chapter 12 (Canonical Perturbation Theory) relies on Chapters 8–10. Some examples in Section 12.3 are based on Sections 3.7–8 (the Kepler Problem).
• Chapter 13 (Continuous Systems & Fields) generally benefits from knowledge of Chapter 7 (Special Relativity), especially in Section 13.5 (Relativistic Field Theory). Section 13.4 (Hamiltonian Field Theory) requires familiarity with Chapter 8.