In addition to the mentioned very good references there is also the book by Shigeru Mukai "An introduction to invariants and moduli" Cambridge University Press, Cambridge, 2003. ISBN: 0-521-80906-1.
Also, there is not much difference between classical invariant theory (as done by the classics themselves) and GIT, they both are about studying quotients in algebraic geometry. If you need to work with explicit coordinates on your quotient then you are doing CIT, if not you are doing GIT.
