Selective transport of liquids is an important process in the energy and environment industry. The increased energy consumption and the demands of clean water and fossil fuels have urged the development of high-performance membrane technologies. Nanoscale channels with the critical size for molecular sieving and atomistically smooth walls for significant boundary slippage are highly promising to balance the tradeoff between permeability and selectivity. In this work, we explore the molecular structures and dynamics of organic solvents and water, which are confined within nanoscale two-dimensional galleries between graphene or graphene oxide sheets. Molecular dynamics simulation results show that the layered order and significant interfacial slippage are universal for all molecular liquids, leading to notable flow enhancement for channels with a width of few nanometers, in the order of ethylene glycol > butanol > ethanol > hexane > toluene > water > acetone. The extracted dependence of permeability, selectivity on the channel width, and properties of molecular liquids clarify the underlying mechanisms of selective mass transport in nanofluidics, which help to understand and control the filtration and separation processes of molecular liquids. The performance of graphene oxide membranes for permeation and filtration is finally discussed based on the calculated flow resistance for pressure-driven flow or molecular diffusivity for diffusive flow, as well as the solubility and wettability of membranes.