Complex analysis book with a view toward Riemann surfaces?

Narasimhan-Nievergelt's Complex Analysis in One Variable is exactly the book you want.

It is completely geometric and will introduce you, starting from scratch, not only to Riemann surfaces but also to the theory or holomorphic functions of several variables, covering spaces, cohomology,...
This unique book emphasizes how little you have to know of the classical function of one complex variable: just the forty pages of Chapter 1, aptly named Elementary Theory of Holomorphic Functions.
A book with a similar philosophy is Analyse Complexe by Dolbeault, he of the Dolbeault cohomology, which has the drawback of being in French (albeit in mathematical French, which is a far cry from Mallarmé or Proust French...)

It is an underappreciated fact, displayed in both these books, that most of the material found in books on complex analysis of one variable is useless for the study of Riemann surfaces and more generally complex manifolds.
For example all the clever computations of real integrals by residue calculus, evaluation of convergence radius of power series, asymptotic methods, Weierstraß products, Schwarz-Christoffel transformations, ... are irrelevant in complex analytic geometry: I challenge anyone to find the slightest trace of these in the work of the recently deceased H. Grauert, arguably the greatest 20th century specialist in the geometry of complex analytic spaces.

I highly recommend Algebraic Curves and Riemann Surfaces by Prof. Rick Miranda.

Jones and Singerman, Complex Functions: An algebraic and geometric viewpoint.