A good introduction to derived categories of coherent sheaves are Caldaru's notes [Derived categories of sheaves: a skimming][1].

As for Bridgeland's work, I would recommend reading his papers directly, using Huybrecht's book as a reference (as mentioned by Francesco above, "Fourier-Mukai transforms in algebraic geometry"). Specifically for Bridgeland stability conditions, I also have some short notes on
[my homepage][2], but again, you should also read his original papers.


  [1]: https://arxiv.org/abs/math/0501094
  [2]: http://www.math.uconn.edu/~bayer/