We use the terahertz (THz) emission spectroscopy to study femtosecond photocurrent dynamics in the prototypical 2D semiconductor, transition metal dichalcogenide MoSe2. We identify several distinct mechanisms producing THz radiation in response to an ultrashort (30 fs) optical excitation in a bilayer (BL) and a multilayer (ML) sample. In the ML, the THz radiation is generated at a picosecond timescale by out-of-plane currents due to the drift of photoexcited charge carriers in the surface electric field. The BL emission is generated by an in-plane shift current. Finally, we observe oscillations at about 23 THz in the emission from the BL sample. We attribute the oscillations to quantum beats between two excitonic states with energetic separation of similar to 100 meV.