[An Invitation to Quantum Cohomology: Kontsevich's Formula for Rational Plane Curves][1] 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.


  [1]: http://books.google.com/books?id=NCNxbJ5_r9UC&printsec=frontcover&dq=An+invitation+to+quantum+cohomology%3A+Kontsevich%2527s+formula&source=bl&ots=XMfn5TPe66&sig=vCcxKAAEl2XgDq5fqtYeFByWUy8&hl=no&ei=WiHLTKnZC8rv4AaHw-DcDA&sa=X&oi=book_result&ct=result&resnum=3&ved=0CCcQ6AEwAg