In living cells, protein-rich condensates can wet the cell membrane and surfaces of membrane-bound organelles. Interestingly, many phase-separating proteins also bind to membranes leading to a molecular layer of bound molecules. Here we investigate how binding to membranes affects wetting, prewetting and surface phase transitions. We derive a thermodynamic theory for a three-dimensional bulk in the presence of a two-dimensional, flat membrane. At phase coexistence, we find that membrane binding facilitates complete wetting and thus lowers the wetting angle. Moreover, below the saturation concentration, binding facilitates the formation of a thick layer at the membrane and thereby shifts the prewetting phase transition far below the saturation concentration. The distinction between bound and unbound molecules near the surface leads to a large variety of surface states and complex surface phase diagrams with a rich topology of phase transitions. Our work suggests that surface phase transitions combined with molecular binding represent a versatile mechanism to control the formation of protein-rich domains at intra-cellular surfaces.