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I think you would profit more from studying the first five sections of these notes by Larry Breen

Notes on 1- and 2-Gerbes, in: Baez J., May J. (eds) Towards Higher Categories. The IMA Volumes in Mathematics and its Applications 152 (2010) pp 193-235, doi:10.1007/978-1-4419-1524-5_5 arXiv:math/0611317

 

Abstract: The aim of these notes is to discuss in an informal manner the construction and some properties of 1- and 2-gerbes. They are for the most part based on the author’s texts [1–4]. Our main goal is to describe the construction which associates to a gerbe or a 2-gerbe the corresponding non-abelian cohomology class.

While it is true that this is all subsumed by Lurie's work, and is also covered by Jardine's work (less general than Lurie's), it is not what you need to read when starting out, if all you want is to understand gerbes, and not $\infty$-stacks that are (higher) gerbes.

Breen's work is a synthesis of work by Debremaeker (in this thesis), Duskin (An outline of non-abelian cohomology in a topos. I. The theory of bouquets and gerbes), Ulbrich (On the correspondence between gerbes and bouquets), and some of his own (and perhaps a couple more that I forgot). Ultimately the roots of the idea go back to Giraud, but I don't recommend reading his book as his cohomology theory is the wrong one (see this answer I once wrote).

I think you would profit more from studying the first five sections of these notes by Larry Breen

Notes on 1- and 2-Gerbes, in: Baez J., May J. (eds) Towards Higher Categories. The IMA Volumes in Mathematics and its Applications 152 (2010) pp 193-235, doi:10.1007/978-1-4419-1524-5_5 arXiv:math/0611317

 

Abstract: The aim of these notes is to discuss in an informal manner the construction and some properties of 1- and 2-gerbes. They are for the most part based on the author’s texts [1–4]. Our main goal is to describe the construction which associates to a gerbe or a 2-gerbe the corresponding non-abelian cohomology class.

While it is true that this is all subsumed by Lurie's work, and is also covered by Jardine's work (less general than Lurie's), it is not what you need to read when starting out, if all you want is to understand gerbes, and not $\infty$-stacks that are (higher) gerbes.

Breen's work is a synthesis of work by Debremaeker (in this thesis), Duskin (An outline of non-abelian cohomology in a topos. I. The theory of bouquets and gerbes), Ulbrich (On the correspondence between gerbes and bouquets), and some of his own (and perhaps a couple more that I forgot). Ultimately the roots of the idea go back to Giraud, but I don't recommend reading his book as his cohomology theory is the wrong one (see this answer I once wrote).

I think you would profit more from studying the first five sections of these notes by Larry Breen

Notes on 1- and 2-Gerbes, in: Baez J., May J. (eds) Towards Higher Categories. The IMA Volumes in Mathematics and its Applications 152 (2010) pp 193-235, doi:10.1007/978-1-4419-1524-5_5 arXiv:math/0611317

Abstract: The aim of these notes is to discuss in an informal manner the construction and some properties of 1- and 2-gerbes. They are for the most part based on the author’s texts [1–4]. Our main goal is to describe the construction which associates to a gerbe or a 2-gerbe the corresponding non-abelian cohomology class.

While it is true that this is all subsumed by Lurie's work, and is also covered by Jardine's work (less general than Lurie's), it is not what you need to read when starting out, if all you want is to understand gerbes, and not $\infty$-stacks that are (higher) gerbes.

Breen's work is a synthesis of work by Debremaeker (in this thesis), Duskin (An outline of non-abelian cohomology in a topos. I. The theory of bouquets and gerbes), Ulbrich (On the correspondence between gerbes and bouquets), and some of his own (and perhaps a couple more that I forgot). Ultimately the roots of the idea go back to Giraud, but I don't recommend reading his book as his cohomology theory is the wrong one (see this answer I once wrote).

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I think you would profit more from studying the first five sections of these notes by Larry Breen

Notes on 1- and 2-Gerbes, in: Baez J., May J. (eds) Towards Higher Categories. The IMA Volumes in Mathematics and its Applications 152 (2010) pp 193-235, doi:10.1007/978-1-4419-1524-5_5 arXiv:math/0611317

Abstract: The aim of these notes is to discuss in an informal manner the construction and some properties of 1- and 2-gerbes. They are for the most part based on the author’s texts [1–4]. Our main goal is to describe the construction which associates to a gerbe or a 2-gerbe the corresponding non-abelian cohomology class.

While it is true that this is all subsumed by Lurie's work, and is also covered by Jardine's work (less general than Lurie's), it is not what you need to read when starting out, if all you want is to understand gerbes, and not $\infty$-stacks that are (higher) gerbes.

Breen's work is a synthesis of work by Debremaeker (in this thesis), Duskin (An outline of non-abelian cohomology in a topos. I. The theory of bouquets and gerbes), Ulbrich (On the correspondence between gerbes and bouquets), and some of his own (and perhaps a couple more that I forgot). Ultimately the roots of the idea go back to Giraud, but I don't recommend reading his book as his cohomology theory is the wrong one (see this answer I once wrote).