I'm no expert on homological algebra, but I have often read that there are many issues with triangulated categories, such as certain maps involving mapping cones failing to be unique, and poor behaviour of the triangulated structure under various constructions. The usual explanation for this that I see is that a triangulated category is in general a "squashed down" stable $(\infty,1)$-category, and the issues come from ignoring the higher cells.
Anyway, as a fan of "abstract nonsense", I am curious if there is a reference that develops homological algebra from the point of view of Quillen model structures and/or stable $(\infty,1)$-category theory?
With regards to my background, I'm pretty comfortable with 1-category theory, but don't have any real background in higher category theory. I'm just starting to read Riehl & Verity's Infinity Category Theory from Scratch. My homotopy theory background is essentially the first two chapters of Emily Riehl's Categorical Homotopy Theory (so basically up to derived functors as Kan extensions), but I can pick that up as I go.
The only reference I know of is in the $\infty$-categorical (specifically, quasicategorical) setting, which is Higher Algebra by Lurie; but if you know what a stable $\infty$-category is, then you almost certainly know about this reference. Though I do have to say that Lurie does give a complete treatment of the general setup you would find in any traditional text on homological algebra, that is, Lurie defines derived categories and functors, and also discusses spectral sequences etc.; it just doesn't discuss any concrete applications in depth, as say Weibel does with e.g. group cohomology.
To read this text you should of course learn about quasicategories. I have added some references:
In a model categorical setting I don't know of any general treatment of homological algebra; in fact, I only know of one source discussing the stable theory from scratch, which is the book by Hovey.