Recent experimental and theoretical studies of biomimetic membrane adhesions [Bruinsma et al., Phys. Rev. E 61, 4253 (2000); Boulbitch et al., Biophys. J. 81, 2743 (2001)] suggested that adhesion mediated by receptor interactions is due to the interplay between membrane undulations and a double-well adhesion potential, and should be a first-order transition. We study the nucleation of membrane adhesion by finding the minimum-energy path on the free energy surface constructed from the bending free energy of the membrane and the double-well adhesion potential. We find a nucleation free energy barrier around 20kBT for adhesion of flexible membranes, which corresponds to fast nucleation kinetics with a time scale of the order of seconds. For cell membranes with a larger bending rigidity due to the actin network, the nucleation barrier is higher and may require active processes such as the reorganization of the cortex network to overcome this barrier. Our scaling analysis suggests that the geometry of the membrane shapes of the adhesion contact is controlled by the adhesion length that is determined by the membrane rigidity, the barrier height, and the length scale of the double-well potential, while the energetics of adhesion is determined by the depths of the adhesion potential. These results are verified by numerical calculations.