We present an optical spectroscopic survey of 24 micron and 1.4 GHz sources, detected in the Spitzer Extragalactic First Look Survey (FLS), using the multi-fiber spectrograph, Hydra, on the WIYN telescope. We have obtained spectra for 772 sources, with flux densities above 0.15 mJy in the infrared, and 0.09 mJy in the radio. The redshifts measured in this survey are mostly in the range 0 < z < 0.4, with a distribution peaking at z = 0.2. Detailed spectral analysis of our sources reveals that the majority are emission-line star-forming galaxies, with star formation rates in the range 0.2-200 Msun/yr. The rates estimated from the H-alpha line fluxes are found to be on average consistent with those derived from the 1.4 GHz luminosities. For these star-forming systems, we find that the 24 micron and 1.4 GHz flux densities follow an infrared-radio correlation, that can be characterized by a value of q24 = 0.83, with a 1-sigma scatter of 0.31. Our WIYN/Hydra database of spectra complements nicely those obtained by the Sloan Digital Sky Survey, in the region at lower redshift, as well as the MMT/Hectospec survey of Papovich et al. (2006), and brings the redshift completeness to 70% for sources brighter than 2 mJy at 24 micron. Applying the classical 1/Vmax method, we derive new 24 micron and 1.4 GHz luminosity functions, using all known redshifts in the FLS. We find evidence for evolution in both the 1.4 GHz and 24 micron luminosity functions in the redshift range 0 < z < 1. The redshift catalog and spectra presented in this paper are available at the Spitzer FLS website.
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