It is hard to give references to learn this material that are approachable for a beginner. He are some of the materials that I used while preparing these notes.

J. Lurie - Higher Topos Theory - the source with most of the technical results one might need while working with oo-categories. The book takes quasi-categories as its working model, however it uses different models for certain important constructions. Chapter 1 is very approachable and gives a good summary of the theory for beginners.

Chapter 1 of J. Lurie - Higher Algebra - chapter develops the analog of abelian categories in the context of oo-categories, i.e. stable oo-categories. This is extremely relevant since in derived algebra geometry most sheaves will take place in stable oo-categories.

C. Rezk - Introduction to Quasicategories: C. Rezk is writing a very nice and self-contained notes developing all operations of category theory fully within the model of quasi-categories. He's approach is closer to Joyal's historical work and is a ''lecture notes'' presentation (he nicely explains what that means in the introduction).

M. Groth - A short course on oo-categories: this reference discusses oo-categories also using the model of quasi-categories it can be taken as an introduction to HTT and certain parts of HA.

Chapter 1 of D. Gaitsgory and N. Rozenblyum - A study in derived algebraic geometry: Volume I: this chapter gives a concise introduction for practical knowledge assuming that one defined the basics of the theory.

E. Riehl and D. Verity - Elements of oo-Category Theory: book embarks in developing the category theory in a mode-independent way starting with the assumption that one has at least one model for the homotopy 2-category of (oo,1)-categories.

A. Mazel-Gee - The Zen of oo-categories: these short notes by Aaron Mazel-Gee give a good intuition, from a model-independet point of view, for what are key features about oo-categories (specially section 5).

J. Bergner - The homotopy theory of (oo,1)-categories: is a very approachable book where one can find careful definitions of many models for oo-categories and proofs or great sketches for why they are equialent. See also this notes by J. Bergner that serve as an introduction to the book.

J. Lurie - Kerodon: this growing online repository has a very user-friendly development of oo-categories and possibly more topics to come. The presentation is structured differently than HTT and it can serve as a good complementary/introduction to reading it.

B. Toën and G. Vezzosi - Homotopical Algebraic Geometry I: this volume develops background material on oo-topoi and stable categories from the point of view of model categories.

J. Lurie - Derived algebraic geometry: this is much shorter and a bit less comprehensive than other texts, but it is slightly more informal which makes for a great introduction to this material.

Chapter 7 of J. Lurie - Higher Algebra: this develops the basics of the theory of affine derived geometry. In fact, the chapter is more general since it treats E_k-algebras and their modules, we will be interested only in the E_oo case. It relies on certain previous sections of the both, but can be approached almost independently.

D. Gaitsgory and N. Rozenblyum - A study in derived algebraic geometry Volumes I and I: Chapter 2 of Volume I gives a good summary of many formal properties of DAG from a functor of points perspective. Chapter 1 of Volume II develops the theory of the cotangent complex in great generality.

J. Lurie - Spectral Algebraic Geometry: this develops the non-affine part of the theory in the most general of contexts. It is technically outside the scope of what we want to understand, but it might contain useful general results and discussion, specially in the introduction to different chapters.

B. Toën and G. Vezzosi - Homotopical Algebraic Geometry II: also using model categories, this volume presents a framework of "homotopical context" that can be used to develop the theory of classical higher stacks (`a la Simpson), derived algebraic geometry over cdgas, or over E_{oo}-rings.

Many people ran seminars and courses on this. I am not following any other course in particular, but I borrowed and learned from some of the following.

B. Toën - Derived Algebraic Geometry: reference gives a great summary of theory with a short introduction to oo-categories included as well as many applications to the construction of moduli spaces, and an introduction to shifted symplectic geometry.

M. Porta ran a couple of seminars on this: you can find very useful material on his page.

P. Safronov also ran a seminar: you can find some notes here.