References on Gerbes

My personal impression is that at least on the level of foundational theory, Higher Topos Theory of Lurie is a good source. I guess this also explain the hard time you feel finding references: Gerbes seat very naturally in the context of sheaves of spaces (in this language this is just a connected sheaf of 1-types!), and I guess that this language has not fully penetrated into standard algebraic geometry texts yet, or any subject which is not modern algebraic topology, actually. However, the situation do get better with time, and I think that gerbes will appear more in texts soon (in particular, they are not out of fashion, just sort of get revised by $\infty$-category theory). For example, I personally almost finished a paper with a whole section for gerbes-based obstruction theory in etale homotopy, so I know there's at least one text on the subject that will be on the archive soon :-)

The book of Giraud is a fundamental reference on the subject, but you have to be used to the language of Grothendieck. A reference more accessible, for example for a differential geometer is the chapter 5 of the book of Brylinski which deals only with commutative gerbes.

J.L Brylinski Loop Spaces, Characteristic Classes and Geometric Quantization.

Urs Schreiber has written a lot on gerbes and their applications to physics: https://ncatlab.org/nlab/show/Urs+Schreiber

See, for instance, the expository works “Differential cohomology in a cohesive ∞-topos” and “Higher prequantum geometry”. Most of his published papers (https://arxiv.org/find/math/1/au:+Schreiber_U/0/1/0/all/0/1) use gerbes in some way.

Almost all of Konrad Waldorf's papers also involve gerbes in some way, and some of them may be more accessible to a beginner, see, e.g., his survey with Christoph Schweigert Gerbes and Lie Groups.