Young, low-mass stars in the solar neighborhood are vital for completing the mass function for nearby, young coeval groups, establishing a more complete census for evolutionary studies, and providing targets for direct-imaging exoplanet and/or disk studies. We present properties derived from high-resolution optical spectra for 336 candidate young nearby, low-mass stars. These include measurements of radial velocities and age diagnostics such as H{alpha} and Li {lambda}6707 equivalent widths. Combining our radial velocities with astrometry from Gaia DR2 (Cat. I/345), we provide full 3D kinematics for the entire sample. We combine the measured spectroscopic youth information with additional age diagnostics (e.g., X-ray and UV fluxes, color-magnitude diagram positions) and kinematics to evaluate potential membership in nearby, young moving groups and associations. We identify 77 objects in our sample as bona fide members of 10 different moving groups, 14 of which are completely new members or have had their group membership reassigned. We also reject 44 previously proposed candidate moving group members. Furthermore, we have newly identified or confirmed the youth of numerous additional stars that do not belong to any currently known group and find 69 comoving systems using Gaia DR2 astrometry. We also find evidence that the Carina association is younger than previously thought, with an age similar to the {beta} Pictoris moving group (~22 Myr).