We present a spectroscopic survey using the MMT Hectospec fiber spectrograph of 24 μm sources selected with the Spitzer Space Telescope in the Spitzer First Look Survey. We report 1296 new redshifts for 24 μm sources, including 599 with fν(24 μm) ≥ 1 mJy. Combined with 291 additional redshifts for sources from the Sloan Digital Sky Survey (SDSS), our observing program was highly efficient and is ~90% complete for i ≤ 21 mag and fν(24 μm) ≥ 1 mJy and 35% complete for i ≤ 20.5 mag and 0.3 mJy ≤ fν(24 μm) < 1.0 mJy. Our Hectospec survey includes 1078 and 168 objects spectroscopically classified as galaxies and QSOs, respectively. Combining the Hectospec and SDSS samples, we find 24 μm-selected galaxies to zgal ≤ 0.98 and QSOs to zQSO ≤ 3.6, with mean redshifts of zgal = 0.27 and zQSO = 1.1. As part of this publication, we include the redshift catalogs and the reduced spectra; these are also available through the NASA/IPAC Infrared Science Archive.