Parallaxes, proper motions, and optical photometry are presented for 51 systems consisting of 37 cool subdwarf and 14 additional high proper motion systems. Thirty-seven systems have parallaxes reported for the first time, 15 of which have proper motions of at least 1"/yr. The sample includes 22 newly identified cool subdwarfs within 100 pc, of which three are within 25 pc, and an additional five subdwarfs from 100 to 160 pc. Two systems-LSR 1610-0040 AB and LHS 440 AB-are close binaries exhibiting clear astrometric perturbations that will ultimately provide important masses for cool subdwarfs. We use the accurate parallaxes and proper motions provided here, combined with additional data from our program and others, to determine that effectively all nearby stars with tangential velocities greater than 200 km/s are subdwarfs. We compare a sample of 167 confirmed cool subdwarfs to nearby main sequence dwarfs and Pleiades members on an observational Hertzsprung-Russell diagram using M_V_ versus (V-K_s_) to map trends of age and metallicity. We find that subdwarfs are clearly separated for spectral types K5-M5, indicating that the low metallicities of subdwarfs set them apart in the H-R diagram for (V-K_s_)=3-6. We then apply the tangential velocity cutoff and the subdwarf region of the H-R diagram to stars with parallaxes from Gaia Data Release 1 and the MEarth Project to identify a total of 29 new nearby subdwarf candidates that fall clearly below the main sequence.