Let C be a smooth complex projective curve of genus g and let C(2) be its second symmetric product. This paper concerns the study of some attempts at extending to C(2) the notion of gonality. In particular, we prove that the degree of irrationality of C(2) is at least g-1 when C is generic, and that the minimum gonality of curves through the generic point of C(2) equals the gonality of C. In order to produce the main results we deal with correspondences on the k-fold symmetric product of C, with some interesting linear subspaces of P^n enjoying a condition of Cayley-Bacharach type, and with monodromy of rational maps. As an application, we also give new bounds on the ample cone of C(2) when C is a generic curve of genus 6 ≤ g ≤ 8.