The reading strategy outlined here is approximately the following:

Start with appendix A.2.

Continue with the overview chapter 1.

Chapter 2 developes the theory of fibrations of simplicial sets.The aims of this are mainly three different concerns:

Establishing the $\infty$-Grothendieck construction: The type of fibrations accomplishing this are left/right fibrations (aka. fibrations in groupoids) and cartesian fibrations (aka. Grothendieck fibrations).

Preparing the Joyal model structure: This is a foundational topic; the fibrant objects of this model structure are precisely $/infty$-categories. The technical vehicle for this are anodyne maps.

Provide a foundations for a theory of $n$-categories, for any $n\le\infty$. For the well definedness of this notion minimal fibrations (a special kind of inner fibrations) are introduced.

Omit chapter 3.

The rest of the book is concerned with constructions which in most cases are proposed in chapter 2. So concentrate on the following important topics: