Microtubules are indispensable for Golgi complex assembly and maintenance, which are integral parts of cytoplasm organization during interphase in mammalian cells. Here, we show that two discrete microtubule subsets drive two distinct, yet simultaneous, stages of Golgi assembly. In addition to the radial centrosomal microtubule array, which positions the Golgi in the centre of the cell, we have identified a role for microtubules that form at the Golgi membranes in a manner dependent on the microtubule regulators CLASPs. These Golgi-derived microtubules draw Golgi ministacks together in tangential fashion and are crucial for establishing continuity and proper morphology of the Golgi complex. We propose that specialized functions of these two microtubule arrays arise from their specific geometries. Further, we demonstrate that directional post-Golgi trafficking and cell migration depend on Golgi-associated CLASPs, suggesting that correct organization of the Golgi complex by microtubules is essential for cell polarization and motility.