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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.