This is the trickiest part. What one needs to know is "how infinity-categories work", but not necessarily how the theory is constructed (whether via quasi-categories, topological categories or complete Segal spaces). Below are some suggestions for the sources where to read up on this.
Dennis Gaitsgory finds reading Section 1 of Lurie's Higher Topos Theory most helpful. The idea is to understand the spirit of things, and then try to apply it yourself.
Dustin Clausen recommends Lurie's infinity-topoi paper.
David Ben-Zvi recommends notes by H. Tanaka (available for download below).
Once somewhat familiar with the language, upon the recommendation of both Dustin and Jacob, it is a good idea to read this paper by Ben-Zvi, Francis, and Nadler just to see how the theory is applied.
As examples of self-taught application of the theory, it may be a good idea to familiarize yourself with Gaitsgory's notes on stacks and quasi-coherent sheaves. Another good reason to study these texts is that they can serve as a background reading for the (technically crucial) Lecture 1.2.
Another good survey of the basic facts is Section 2.1 of Toen's paper 'Derived Algebraic Geometry' and Section 1 of Chern character paper by Toen and Vezzosi.
Going back to the general theory, Sam Raskin suggests the survey by Omar Camarena.
Returning to the foundations, David Nadler suggests the following three papers:
Bergner, A survey of (∞,1)-categories
Bergner, Models for (∞,n)-categories and the cobordism hypothesis
Barwick and Schommer-Pries, On the Unicity of the Homotopy Theory of Higher Categories