While the star formation rates and morphologies of galaxies have long been known to correlate with their local environment, the process by which these correlations are generated is not well understood. Galaxy groups are thought to play an important role in shaping the physical properties of galaxies before entering massive clusters at low redshift, and transformations of satellite galaxies likely dominate the buildup of local environmental correlations. To illuminate the physical processes that shape galaxy evolution in dense environments, we study a sample of 116 X-ray selected galaxy groups at z=0.2-1 with halo masses of 10^13-10^14 M_sun and centroids determined with weak lensing. We analyze morphologies based on HST imaging and colors determined from 31 photometric bands for a stellar mass-limited population of 923 satellite galaxies and a comparison sample of 16644 field galaxies. Controlling for variations in stellar mass across environments, we find significant trends in the colors and morphologies of satellite galaxies with group-centric distance and across cosmic time. Specifically at low stellar mass (log(M_stellar/M_sun) = 9.8-10.3), the fraction of disk-dominated star-forming galaxies declines from >50% among field galaxies to <20% among satellites near the centers of groups. This decline is accompanied by a rise in quenched galaxies with intermediate bulge+disk morphologies, and only a weak increase in red bulge-dominated systems. These results show that both color and morphology are influenced by a galaxys location within a group halo. We suggest that strangulation and disk fading alone are insufficient to explain the observed morphological dependence on environment, and that galaxy mergers or close tidal encounters must play a role in building up the population of quenched galaxies with bulges seen in dense environments at low redshift.