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We present the first results from the deep and wide 5 GHz radio observations of the Great Observatories Origins Deep Survey (GOODS)-North (σ = 3.5 μJy beam−1, synthesized beam size θ = 1.″47 × 1.″42, and 52 sources over 109 arcmin2) and GOODS-South (σ = 3.0 μJy beam−1, θ = 0.″98 × 0.″45, and 88 sources over 190 arcmin2) fields using the Karl G. Jansky Very Large Array. We derive radio spectral indices α between 1.4 and 5 GHz using the beam-matched images and show that the overall spectral index distribution is broad even when the measured noise and flux bias are considered. We also find a clustering of faint radio sources around α = 0.8, but only within S5 GHz < 150 μJy. We demonstrate that the correct radio spectral index is important for deriving accurate rest-frame radio power and analyzing the radio–FIR correlation, and adopting a single value of α = 0.8 leads to a significant scatter and a strong bias in the analysis of the radio–FIR correlation, resulting from the broad and asymmetric spectral index distribution. When characterized by specific star formation rates, the starburst population (58%) dominates the 5 GHz radio source population, and the quiescent galaxy population (30%) follows a distinct trend in spectral index distribution and the radio–FIR correlation. Lastly, we offer suggestions on sensitivity and angular resolution for future ultra-deep surveys designed to trace the cosmic history of star formation and AGN activity using radio continuum as a probe. [ABSTRACT FROM AUTHOR]