We present a spectroscopic catalog of around 1600 high proper motion ( > 0.4"/yr) M dwarfs and M subdwarfs, collected from medium-resolution observations at the MDM observatory. Since a high proper motion sample is over-represented with high-velocity stars which are sampled over a larger volume as compared to low-velocity stars, our catalog includes a significant number of stars belonging to old Galactic populations, i.e., thick disk and halo. Using synthetic-model fitting, we have obtained the metallicity, alpha-element abundances, effective temperature and surface gravity of our stars. We have also been able to construct the H-R diagram of these stars, based on the accurate parallaxes from Gaia DR2. This diagram shows stars with different metallicity ranges fall into clearly distinct loci which can be used to develop photometric metallicity calibrations, in particular, for metal-poor M subdwarfs. In addition, we have identified co-moving pairs in our catalog which can provide excellent targets for future high-resolution surveys of M dwarfs.