A good introduction to derived categories of coherent sheaves are Caldaru's notes Derived categories of sheaves: a skimmingDerived categories of sheaves: a skimming.
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, but again, you should also read his original papers.