Derived algebraic geometry: how to reach research level math?

I propose the following plan, assuming a basic background in scheme theory and algebraic topology. I assume that you are interested in derived algebraic geometry from the point of view of applications in algebraic geometry. (If you are interested in applications to topology, you should replace part 2) of the plan by Lurie's Higher algebra.) The plan is based on what worked best for myself, and it's certainly possible that you may prefer to jump into Higher Topos Theory as Yonatan suggested.

0) First of all, make sure you have a solid grounding in basic category theory. For this, read the first two chapters of the excellent lecture notes of Schapira. I would strongly recommend reading chapters 3 and 4 as well, but these can be skipped for now.

Then read chapters I and II of Gabriel-Zisman, Calculus of fractions and homotopy theory, to learn about the theory of localization of categories.

1) The next step is to learn the basics of abstract homotopy theory.

I recommend working through Cisinski's notes. This will take you through simplicial sets, model categories, a beautiful construction of the Quillen and Joyal model structures (which present $\infty$-groupoids and $\infty$-categories, respectively), and the fundamental constructions of $\infty$-category theory (functor categories, homotopy (co)limits, fibred categories, prestacks, etc.).

Supplement the section "Catégories de modèles" with chapter I of Quillen's lecture notes Homotopical algebra.

Then read about stable $\infty$-categories and symmetric monoidal $\infty$-categories in these notes from a mini-course by Cisinski. (By the way, these ones are in English and also summarize very briefly some of the material from the longer course notes). These notes are very brief, so you will have to supplement them with the notes of Joyal. It may also be helpful to have a look at the first chapter of Lurie's Higher algebra and the notes of Moritz Groth.

2) At this point you are ready to learn some derived commutative algebra:

Read lecture 4 of part II of Moerdijk-Toen, Simplicial Methods for Operads and Algebraic Geometry together with section 3 of Lurie's thesis. Supplement this with section 2.2.2 of Toen-Vezzosi's HAG II, referring to chapter 1.2 when necessary. This material is at the heart of derived algebraic geometry: the cotangent complex, infinitesimal extensions, Postnikov towers of simplicial commutative rings, etc.

Other helpful things to look at are Schwede's Diplomarbeit and Quillen's Homology of commutative rings.

3) Before learning about derived stacks, I would strongly recommend working through these notes of Toen about classical algebraic stacks, from a homotopy theoretic perspective. There are also these notes of Preygel. This will make it a lot easier to understand what comes next.

Then, read Lurie's On $\infty$-topoi. It will be helpful to consult sections 15-20 of Cisinski's Bourbaki talk, section 40 of Joyal's notes on quasi-categories, and Rezk's notes. For a summary of this material, see lecture 2 of Moerdijk-Toen.

4) Finally, read about derived stacks in lecture 5 of Moerdijk-Toen and section 5 of Lurie's thesis. Again, chapters 1.3, 1.4, and 2.2 of HAG II will be very helpful references. See also Gaitsgory's notes (he works with commutative connective dg-algebras instead of simplicial commutative rings, but this makes little difference). His notes on quasi-coherent sheaves in DAG are also very good.

5) At this point, you know the definitions of objects in derived algebraic geometry. To get some experience working with them, I would recommend reading some of the following papers:

• Antieau-Gepner, Brauer groups and étale cohomology in derived algebraic geometry, arXiv:1210.0290
• Bhatt, p-adic derived de Rham cohomology, arXiv:1204.6560.
• Bhatt-Scholze, Projectivity of the Witt vector affine Grassmannian, arXiv:1507.06490.
• Gaitsgory-Rozenblyum, A study in derived algebraic geometry, link
• Kerz-Strunk-Tamme, Algebraic K-theory and descent for blow-ups, arXiv:1611.08466.
• Toen, Derived Azumaya algebras and generators for twisted derived categories, arXiv:1002.2599.
• Toen, Proper lci morphisms preserve perfect complexes, arXiv:1210.2827.
• Toen-Vaquie, Moduli of objects in dg-categories, arXiv:math/0503269.

In my opinion the best foundations to any modern topic in homotopy theory, and derived algebraic geometry in particular, is "Higher topos theory" of Lurie. The scope covers all the required ($\infty$-)categorical framework, and every chapter starts with a very conceptual motivation. In addition, the book also contains appendices which explain classical material (such as model categories) in a very readable way. You might find in the beginning some proofs which involve technical combinatorics of simplices. Don't be discouraged. Feeling comfortable with simplices is essential and this requires working out some details. The proofs in the book do become increasingly conceptual with each chapter, as the concepts themselves get built and acquire depth.

I'm going to take a dissenting view, here. I think the best way to assimilate concepts in derived algebraic geometry (for finite fields, $\mathbb{R}$ or $\mathbb{C}$), is to understand where and why they are used. Then, work backwards when the need arises. Personally, I found it formidable to read through any section of Toen-Vezzosi's homotopical algebraic geometry series straight through. I'd first recommend reading and understanding the content of Vezzosi's AMS notice, here: https://www.ams.org/notices/201107/rtx110700955p.pdf. Once you begin digesting the need for replacing the source category for Grothendieck's functor of points approach to algebraic geometry with derived commutative algebras, browse through the literature and find instances where this becomes necessary. From my perspective, the most striking application is here: https://arxiv.org/pdf/1102.1150.pdf, where one sees (sloppily speaking here), that even replacing the source category with truncated derived objects goes a very long way in recovering classical results. Feel free to let me know if you'd like me to explicate further.