An Invitation to Quantum Cohomology: Kontsevich's Formula for Rational Plane Curves by Kock and Vainsencher is a really nice introductory book if you also like enumerative problems. The first part of the book gives a nice introduction to moduli problems with special emphasis on the moduli space of pointed stable curves, $M_{g,n}$. There is also a neat proof of Kontsevich's formula and a chapter on Gromov-Witten invariants.