Strong gravitational lenses are now being routinely discovered in wide-field surveys at (sub)millimeter wavelengths. We present Submillimeter Array (SMA) high-spatial resolution imaging and Gemini-South and Multiple Mirror Telescope optical spectroscopy of strong lens candidates discovered in the two widest extragalactic surveys conducted by the Herschel Space Observatory: the Herschel-Astrophysical Terahertz Large Area Survey (H-ATLAS) and the Herschel Multi-tiered Extragalactic Survey (HerMES). From a sample of 30 Herschel sources with S_500>100 mJy, 21 are strongly lensed (multiply imaged), 4 are moderately lensed (singly imaged), and the remainder require additional data to determine their lensing status. We apply a visibility-plane lens modeling technique to the SMA data to recover information about the masses of the lenses as well as the intrinsic (i.e., unlensed) sizes (r_half) and far-infrared luminosities (L_FIR) of the lensed submillimeter galaxies (SMGs). The sample of lenses comprises primarily isolated massive galaxies, but includes some groups and clusters as well. Several of the lenses are located at z_lens>0.7, a redshift regime that is inaccessible to lens searches based on Sloan Digital Sky Survey spectroscopy. The lensed SMGs are amplified by factors that are significantly below statistical model predictions given the 500um flux densities of our sample. We speculate that this may reflect a deficiency in our understanding of the intrinsic sizes and luminosities of the brightest SMGs. The lensed SMGs span nearly one decade in L_FIR (median L_FIR=7.9x10^12 L_sun) and two decades in FIR luminosity surface density (median Sigma_FIR=6.0x10^11 L_sun kpc^-2). The strong lenses in this sample and others identified via (sub-)mm surveys will provide a wealth of information regarding the astrophysics of galaxy formation and evolution over a wide range in redshift. ; Comment: 28 pages, 7 tables, 9 figures. Accepted to ApJ; revised following minor comments from the referee and the community at large