Unfortunately, there is no real textbook on DQ around. One has Fedosov's book on his construction of star products including a detailed exposition of his index theorem.
There is a chapter on DQ in the recent Poisson geometry book by
Laurent-Gengoux, Pichereau, Vanhaecke.
In the conference proceedings of the PQR2003 by Gutt, Rawnsley, and Sternheimer one finds some introductory texts, too.
Concerning the formality theorem of Kontsevich, one has the recent booklet by Esposito, which explains nicely the context (but does not contain the proof of the theorem)
You can also find lecture notes by Simone Gutt "Variations on DQ" or so, they should be on the arXiv somewhere, or on her homepage (?)
And, yes, as Igor mentioned, if you're not too afraid of german, then there is my german textbook on Poisson geometry and DQ with quite a bit of details.
This post imported from StackExchange MathOverflow at 2017-09-17 21:04 (UTC), posted by SE-user Stefan Waldmann