We present the results of SPT-GMOS, a spectroscopic survey with the Gemini Multi-Object Spectrograph (GMOS) on Gemini South. The targets of SPT-GMOS are galaxy clusters identified in the SPT-SZ survey, a millimeter-wave survey of 2500deg^2^ of the southern sky using the South Pole Telescope (SPT). Multi-object spectroscopic observations of 62 SPT-selected galaxy clusters were performed between 2011 January and 2015 December, yielding spectra with radial velocity measurements for 2595 sources. We identify 2243 of these sources as galaxies, and 352 as stars. Of the galaxies, we identify 1579 as members of SPT-SZ galaxy clusters. The primary goal of these observations was to obtain spectra of cluster member galaxies to estimate cluster redshifts and velocity dispersions. We describe the full spectroscopic data set and resulting data products, including galaxy redshifts, cluster redshifts, and velocity dispersions, and measurements of several well-known spectral indices for each galaxy: the equivalent width, W, of [OII]{lambda}{lambda}3727, 3729 and H-{delta}, and the 4000{AA} break strength, D4000. We use the spectral indices to classify galaxies by spectral type (i.e., passive, post-starburst, star-forming), and we match the spectra against photometric catalogs to characterize spectroscopically observed cluster members as a function of brightness (relative to m*). Finally, we report several new measurements of redshifts for ten bright, strongly lensed background galaxies in the cores of eight galaxy clusters. Combining the SPT-GMOS data set with previous spectroscopic follow-up of SPT-SZ galaxy clusters results in spectroscopic measurements for >100 clusters, or ~20% of the full SPT-SZ sample.