A good place to start is :
- Larry Breen. Notes on 1-and 2-gerbes. 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
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 and there you will find a reference to Toby Bartels, Higher gauge theory I: 2-Bundles, amongst lots of other useful sources. Which ones fit best with you is difficult to guess!!!
