What would the minimal prerequisites be to tackle Illusie's (first) book on the cotangent complex?

I guess my goal, other than the moral and professional obligation to learn this 'classical' piece of maths, would be to understand perfect obstruction theories and how these come from 'pulling back structure from the associated derived moduli problem'.

References to where to learn such prerequisites are very much appreciated! (e.g. an introduction to topoi or simplicial sets for the algebro-geometric minded...)