Motivated by recent observation of nematicity in moiré systems, we study three different orbital orders that potentially can happen in moiré systems: (1) the nematic order, (2) the valley polarization, and (3) the "compass order." Each order parameter spontaneously breaks part of the spatial symmetries of the system. We explore physics caused by the quantum fluctuations close to the order-disorder transition of these order parameters. Especially, we recognize that the symmetry of the moiré systems leads to a crucial difference of the effective theory describing the nematic order from the standard Hertz-Millis formalism. We demonstrate that this key difference may lead to a special non-Fermi liquid behavior near the order-disorder nematic transition, different from the standard non-Fermi liquid behavior usually expected when a Fermi surface is coupled to the critical fluctuations of orbital orders. We also discuss the interplay of the three order parameters and the possible rich phase diagram at finite temperature. Within the three orbital orders, the valley polarization and the compass order likely strongly compete with the superconductor. © 2020 American Physical Society.