So, due to the existence of isothermal coordinates, all 2-manifolds are conformally flat. The consequences of this are a bit confusing to me- this means one can conformally map, for instance, the sphere to some surface with identically zero curvature everywhere. What does this surface look like? Is it an infinitely large sphere? (Arbitrarily large radius would imply arbitrarily small curvature, yes?)
Not the entire sphere (which the Gauss-Bonnet theorem would not let you, as Ryan Budney said). But you can conformally map the sphere minus a point onto a plane: this is precisely what the stereographic projection does. The sphere minus the North Pole is mapped onto the plane, as seen in the image below (source).
A closed surface of genus $\ge 2$ can be built from a polygon in the hyperbolic plane by identifying certain sides. Since the hyperbolic polygon is conformally equivalent to a flat one (via the identity map), we again get a conformal map between a flat surface and almost all of the negatively curved surface.