Riemann surfaces is a very standard topics in math, then you can find a lot of books talking about Riemann surfaces under different point of views.
I can suggest you:
-Riemann Surfaces - S.Donaldson,
-Riemann Surfaces - Farkas and Kra,
-Algebraic curves and Riemann surfaces - R.Miranda
-Lectures on Riemann Surfaces - Otto Forster
Donaldson's book is more difficult with respect to others t, and he use a lot of basic algebraic geometry. I have read Forster's book and have been pretty impressed by it. Another excellent analytic monograph from this point of view is the Princeton lecture notes on Riemann surfaces by Robert Gunning, which is also a good place to learn sheaf theory. His main result is that all compact complex one manifolds occur as the Riemann surface of an algebraic curve. Miranda's book contains more study of the geometry of algebraic curves.
Riemann himself, as I recall, took an intermediate view, showing the equivalence of the categories of (irreducible) algebraic curves with that of (connected) compact complex manifolds equipped with a finite holomorphic map to P^1. Another extremely nice book, a little more advanced than Miranda, is the China notes on algebraic curves by Phillip Griffiths. Mumford's book Complex projective varieties I, also has a terrific chapter on curves from the complex analytic point of view.
After you learn the basics, the book of Arbarello, Cornalba, Griffiths, Harris, is just amazing. Of course Riemann's thesis and followup paper on theory of abelian functions is rather incredible as well.