A nice source is "Recursive aspects of descriptive set theory" by Mansfield and Galen Weitkamp, as mentioned by Yu in a comment. A problem with it is that it leaves out all the details of admissibility and its relatives. I also find it a bit more elementary than one would want, but it is a good initial source.
An excellent, encyclopedic reference is a handwritten manuscript by Alain Louveau. I do not know if Alain plans to turn this into a book, but it is very nice and the only source I know for some results. I imagine he may send you a copy if you email him directly.
Alekos Kechris and myself have been working on a book, covering what would be roughly a first semester graduate course on the subject. I am sorry to say that it has taken longer than it should, and it is entirely my fault, as suddenly there were too many things needing my attention at once, and I had to set this apart for a while. But there is a decent working draft, and I expect we should be completing it soonish.
(Once one is past this material, Moschovakis's book is certainly the place to go if one is interested in determinacy and related topics.)
Something that quickly becomes apparent is that a good working understanding of recursion theory is very helpful. Higher recursion theory may help as well, especially if one is interested in more advanced topics (such as applications to the theory of determinacy). Sacks's classical book is a very good introduction. You may also find useful the really nice draft by Chong and Yu, "Recursion Theory: From a generalized point of view", which you can find in Yu's page. (Let me reiterate that it is important to know higher recursion theory if one has interest in determinacy. Several results in the Cabal volumes were established in the language of higher recursion theory, and this shouldn't be an obstacle to understanding them.)
If you want to get started right away, I suggest you read the paper by Martin and Kechris, "Infinite games and effective descriptive set theory", Part 4 of "Analytic sets" by C.A. Rogers et al., Academic Press 1980. Developed from lectures given at the London Mathematical Society Instructional Conference on Analytic Sets, University College London, July 1978. 403-470.