The study of charge and spin transport through semiconductor quantum dots is experiencing a renaissance due to recent advances in nano-fabrication and the realization of quantum dots as candidates for quantum computing. In this work, we combine atomistic electronic structure calculations with quantum master equation methods to study the transport of electrons and holes through strongly confined quantum dots coupled to two leads with a voltage bias. We find that a competition between the energy spacing between the two lowest quasiparticle energy levels and the strength of the exchange interaction determines the spin states of the lowest two quasiparticle energy levels. Specifically, the low density of electron states results in a spin singlet being the lowest energy two-electron state whereas, in contrast, the high density of states and significant exchange interaction results in a spin triplet being the lowest energy two-hole state. The exchange interaction is also responsible for spin blockades in transport properties, which could persist up to temperatures as high as 77K for strongly confined colloidal quantum dots from our calculations. Lastly, we relate these findings to the preparation and manipulation of singlet and triplet spin qubit states in quantum dots using voltage biases.