I am reading the three texts on condensed mathematics by Scholze and Clausen. I also interested in paper "A $p$-adic 6-functor formalism in rigid-analytic geometry" by Lucas Mann.

To advance in the texts I will have to learn about derived categories and later about $\infty$-categories. In these texts the authors treat $\mathcal{D(A)}$ as a $\infty$-category.

Is there a text on derived categories using $\infty$-categories rather than just triangulated categories (that don't require me to study the entirety of higher topos theory to read it)? It is necessary for me to know the construction using triangulated categories first? How much of $\infty$-categories is necessary to read the three condensed texts? What is a good reference for it?