Using first-principles calculations, we investigate the electronic properties of the two-dimensional GaX/MX2 (M = Mo, W; X = S, Se, Te) heterostructures. Orbital hybridization between GaX and MX2 is found to result in Rashba splitting at the valence-band edge around the Γ point, which grows for increasing strength of the spin-orbit coupling in the p orbitals of the chalcogenide atoms. The location of the valence-band maximum in the Brillouin zone can be tuned by strain and application of an out-of-plane electric field. The coexistence of Rashba splitting (in-plane spin direction) and band splitting at the K and K′ valleys (out-of-plane spin direction) makes GaX/MX2 heterostructures interesting for spintronics and valleytronics. They are promising candidates for two-dimensional spin-field-effect transistors and spin-valley Hall effect devices. Our findings shed light on the spin-valley coupling in van der Waals heterostructures.