Controlling thermodynamic cycles to minimize the dissipated heat is a longstanding goal in thermodynamics, and more recently, a central challenge in stochastic thermodynamics for nanoscale systems. Here, we introduce a theoretical and computational framework for optimizing nonequilibrium control protocols that can transform a system between two distributions in a minimally dissipative fashion. These protocols optimally transport a system along a Wasserstein geodesic, paths through the space of probability distributions that minimize the dissipative cost of a transformation. Furthermore, we show that the thermodynamic metric – determined via a linear response approach – can be directly derived from a more general formulation based on optimal transport distances, thus providing a unified perspective on thermodynamic geometries. We investigate this unified geometric framework in two model systems and observe that our procedure for optimizing control protocols is robust beyond linear response.