We investigate two-dimensional hole transport in GaSb quantum wells at cryogenic temperatures using gate-tunable devices. Measurements probing the valence band structure of GaSb unveil a significant spin splitting of the ground subband induced by spin-orbit coupling. We characterize the carrier densities, effective masses and quantum scattering times of these spin-split subbands and find that the results are in agreement with band structure calculations. Additionally, we study the weak anti-localization correction to the conductivity present around zero magnetic field and obtain information on the phase coherence. These results establish GaSb quantum wells as a platform for two-dimensional hole physics and lay the foundations for future experiments in this system.