After reading the first two answers to this question, I've become interested in understanding the concept of (co)tangent complex as a way to get some intuition about homotopical algebra, being somewhat more used to the algebro-geometric framework than to the algebro-topological one. More specifically, I'd like to understand this concept in order 'to do basic geometry - this time calculus - on a singular variety', as stated in the first answer (whatever this means), but not 'mechanically', instead trying always to keep an organizing point of view like the one described in the second answer. Also, I'd like to do so following the shortest possible path from basic algebraic geometry and basic category theory directly to the subject matter, with the smallest possible amount of detours, but comprehensively including all the needed basics. (I've studied some scheme theory and homological algebra before, including derived categories, and also ventured a little bit more deeply in the categorical world, but never dealt professionally with these topics and will have to recall a lot before being sufficiently at ease with them.)
In this context, what I'm looking for is a double list of topics, one from algebraic geometry and the other from category theory, both ordered by degrees of complexity, designed to be studied in a parallel manner, showing the highest possible level of correspondence since the very beginning, and if possible accompanied with the most up-to-date literature available for this purpose. I'd be very grateful if someone would spend some time thinking about this and writing a nice answer.