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.