We consider the Hopf monoid of convex geometries, which admits partial orders as a Hopf submonoid, and investigate combinatorial invariants constructed from characters on them. Each invariant consists of a pair: a polynomial and a more general quasisymmetric function. For partial orders, we obtain the order polynomial of Stanley and the enriched order polynomial of Stembridge. For convex geometries, we obtain polynomials first introduced by Edelman–Jamison and Billera–Hsiao–Provan, which generalized the order and the enriched order polynomials. We obtain reciprocity results satisfied by these polynomials from the perspective of characters in a unified manner. We also describe the coefficients of the quasisymmetric invariants as enumerating faces on certain simplicial complexes. These include the barycentric subdivision of the CW-spheres of convex geometries introduced by Billera, Hsiao, and Provan. For quasisymmetric invariants, we consider the associated $ab$- and $cd$-indices. We discuss an equivalent condition for convex geometries to be supersolvable, which allows us to describe the coefficients of these indices geometrically.