We present a comprehensive examination of optical pumping of spins in individual GaAs quantum dots as we change the charge from positive to neutral to negative using a Schottky diode. We observe that photoluminescence polarization memory has the same sign as the net charge of the dot. Optical pumping of ground state electron spins enhances this effect, as demonstrated through the first measurements of the Hanle effect on an individual quantum dot. With the Overhauser effect in a high longitudinal magnetic field, we demonstrate efficient optical pumping of the quantum dots nuclear spins for all three charge states.