We investigate the effects of three-body and backflow correlations on ground-state properties of the two-dimensional electron gas by both variational and fixed-node Green's function Monte Carlo methods. It is found that the backflow effect is dominant over the three-body effect at high density (rs~1) while they are of equal importance at the lowest density considered (rs~20). With these correlations, we find significant improvements in both variational and fixed-node energies over the Slater-Jastrow results which consider only two-body correlation. The effects are comparable to those in bulk 3He. We present an analytic expression for the correlation energy of the two-dimensional electron gas as a function of the density.