A good place to start is : 

* Larry Breen. [Notes on 1-and 2-gerbes](https://doi.org/10.1007/978-1-4419-1524-5_5). In J. Baez and J. May, editors, Towards Higher Categories, volume 152 of The IMA Volumes in Mathematics and its Applications, pages 193–235. Springer, 2009. [arXiv:math/0611317](https://arxiv.org/abs/math/0611317)

There are lots of other sources and various different approaches to the subject, but I have found Breen's notes very useful. Which approach suits you, of course, depends on your interests and background.  

There are many other more recent treatments.

Remember that principal G bundles are also called G-torsors so search for 2-torsors. There are various papers with 2-torsors in their title, but note there are various levels of generalisation.  For further information try the n-Lab page: [principal 2-bundle](https://ncatlab.org/nlab/show/principal+2-bundle) and there you will find a reference to Toby Bartels, [Higher gauge theory I: 2-Bundles](https://arxiv.org/abs/math/0410328), amongst lots of other useful sources. Which ones fit best with you is difficult to guess!!!