The GALEX Nearby Young Star Search (GALNYSS) yielded the identification of more than 2000 late-type stars that, based on their ultraviolet and infrared colors and pre-Gaia proper motions, are potentially of age<200 Myr and lie within ~120 pc of Earth. We present the results of a campaign of medium- and high-resolution optical spectroscopy of 471 GALNYSS stars aimed at confirming their youth and their potential membership in nearby young stellar moving groups. We present radial velocity (RV), Li absorption, and H-alpha emission measurements for these spectroscopically observed GALNYSS stars, and assess their Li absorption and optical emission-line properties and infrared excesses. Our RV measurements are combined with literature and Gaia DR3 RV measurements and Gaia DR3 astrometry and photometry to obtain the spatiokinematics and color-magnitude positions of GALNYSS stars. We use these results to assess membership in the TW Hya, Tuc-Hor, Carina, Columba, and Argus Associations and the β Pic and AB Dor moving groups. We have identified 132 stars as candidate members of these seven groups; roughly half of these candidates are newly identified on the basis of data presented here. At least one-third of the 132 candidates are spectroscopic and/or photometric binaries and/or have comoving (visual) binary companions in Gaia DR3. The contingent of young, low-mass stars in the solar vicinity we identify here should provide excellent subjects for future direct imaging exoplanet surveys and studies of the early evolution of low-mass stars and their planetary progeny.