After its formation, the Moon is widely believed to have possessed a deep, global magma ocean. As it cooled, an anorthositic crust formed, floating atop this magma ocean, and acting as an insulating blanket. As well as forming the Moon, the Moon-forming giant impact also released more than a lunar mass of debris into heliocentric orbit. Re-impacting debris subjected the newly formed Moon to an extremely intense bombardment. We have conducted a suite of impact simulations for a range of conditions representative of this early period. We find that impact outcomes can be divided into four regimes, and construct scaling relations for the transitions between these regimes and size of impact features. Exposure of liquid magma to the surface is generally more efficient than previously assumed, implying significant shortening of the solidification time of the Lunar Magma Ocean. Comparison with work on icy satellites also suggests that penetration of a solid crust overlying liquid is a relatively universal process with weak dependence on target material properties.