We are carrying out a large ancillary program with the Sloan Digital Sky Survey, SDSS-III, using the fiber-fed multi-object near-infrared APOGEE spectrograph, to obtain high-resolution H-band spectra of more than 1200 M dwarfs. These observations will be used to measure spectroscopic rotational velocities, radial velocities, physical stellar parameters, and variability of the target stars. Here, we describe the target selection for this survey, as well as results from the first year of scientific observations based on spectra that will be publicly available in the SDSS-III DR10 data release. As part of this paper we present radial velocities and rotational velocities of over 200 M dwarfs, with a vsin i precision of ~2 km s–1 and a measurement floor at vsin i = 4 km s–1. This survey significantly increases the number of M dwarfs studied for rotational velocities and radial velocity variability (at ~100-200 m s–1), and will inform and advance the target selection for planned radial velocity and photometric searches for low-mass exoplanets around M dwarfs, such as the Habitable Zone Planet Finder, CARMENES, and TESS. Multiple epochs of radial velocity observations enable us to identify short period binaries, and adaptive optics imaging of a subset of stars enables the detection of possible stellar companions at larger separations. The high-resolution APOGEE spectra, covering the entire H band, provide the opportunity to measure physical stellar parameters such as effective temperatures and metallicities for many of these stars. At the culmination of this survey, we will have obtained multi-epoch spectra and radial velocities for over 1400 stars spanning the spectral range M0-L0, providing the largest set of near-infrared M dwarf spectra at high resolution, and more than doubling the number of known spectroscopic vsin i values for M dwarfs. Furthermore, by modeling telluric lines to correct for small instrumental radial velocity shifts, we hope to achieve a relative velocity precision floor of 50 m s–1 for bright M dwarfs. With three or more epochs, this precision is adequate to detect substellar companions, including giant planets with short orbital periods, and flag them for higher-cadence followup. We present preliminary, and promising, results of this telluric modeling technique in this paper.