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 f _ { \nu } ( 24 \micron ) \geq 1 mJy . Combined with 291 additional redshifts for sources from the Sloan Digital Sky Survey ( SDSS ) , our observing program was highly efficient and is \sim 90 % complete for \hbox { $i$ } \leq 21 mag and f _ { \nu } ( 24 \micron ) \geq 1 mJy , and is 35 % complete for \hbox { $i$ } \leq 20.5 mag and 0.3 mJy \leq f _ { \nu } ( 24 \micron ) < 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 z _ { \mathrm { gal } } \leq 0.98 and QSOs to z _ { \mathrm { QSO } } \leq 3.6 , with mean redshifts of \langle z _ { \mathrm { gal } } \rangle =0.27 and \langle z _ { \mathrm { QSO } } \rangle =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 . http : //irsa.ipac.caltech.edu/