A lot of the material from the Handbook chapter (at least concerning $L$) is also covered in Ralf Schindler's book _Set Theory: Exploring Independence and Truth_ , with the added benefit that more details are worked out and the book covers all the prerequisites. For a fancier treatment that covers more complicated inner models (and also does things like $\Sigma^*$ theory) perhaps Zeman's _Inner Models and Large Cardinals_ might work.