Reference in Riemann Surfaces

I recommend "Lectures on Riemann Surfaces" by Forster.

I strongly reccomend "Algebraic Curves and Riemann Surfaces" by Rick Miranda. I found the book very clearly written, very pedagogical and an excellent reference. It also features quite a lot of exercises, at the end of each subsection, which will help you to get acquainted with the subject.

It depends partly what you are more interested in, geometry or analysis. There are two relevant categories: compact complex manifolds of dimension one (and holomorphic maps), and algebraic complex curves (and rational maps). The approach in the wonderful book of Miranda is to consider the functor from algebraic curves to compact complex one manifolds, although he never fully proves it is well defined. The more analytic approach is to begin with compact complex one manifolds and prove they are all representable as algebraic curves. This is probably the approach of Forster. 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.

Thanks to Georges Elencwajg for significant corrections to this answer.