Depinning and nonequilibrium transitions within sliding states in systems driven over quenched disorder arise across a wide spectrum of size scales ranging from atomic friction at the nanoscale, flux motion in type II superconductors at the mesoscale, colloidal motion in disordered media at the microscale, and plate tectonics at geological length scales. Here we show that active matter or self-propelled particles interacting with quenched disorder under an external drive represents a class of system that can also exhibit pinning-depinning phenomena, plastic flow phases, and nonequilibrium sliding transitions that are correlated with distinct morphologies and velocity-force curve signatures. When interactions with the substrate are strong, a homogeneous pinned liquid phase forms that depins plastically into a uniform disordered phase and then dynamically transitions first into a moving stripe coexisting with a pinned liquid and then into a moving phase-separated state at higher drives. We numerically map the resulting dynamical phase diagrams as a function of external drive, substrate interaction strength, and self-propulsion correlation length. These phases can be observed for active matter moving through random disorder. Our results indicate that intrinsically nonequilibrium systems can exhibit additional nonequilibrium transitions when subjected to an external drive.