The coherent optical injection and temporal decay of spin and charge currents in semiconductor heterostructures is described microscopically, including excitonic effects, many-body Coulomb correlations, and the carrier LO-phonon coupling on the second-order Born-Markov level, as well as nonperturbative light-field-induced intraband and interband excitations. A nonmonotonic dependence of the currents on the intensities of the laser beams is predicted. Enhanced damping of the spin current relative to the charge current is obtained as a consequence of Coulomb scattering.