We present intermediate-resolution optical spectrophotometry of 65 galaxies obtained in support of the Spitzer Infrared Nearby Galaxies Survey (SINGS). For each galaxy we obtain a nuclear, circumnuclear, and semi-integrated optical spectrum designed to coincide spatially with mid- and far-infrared spectroscopy from the Spitzer Space Telescope. We make the reduced, spectrophotometrically calibrated one-dimensional spectra, as well as measurements of the fluxes and equivalent widths of the strong nebular emission lines, publically available. We use optical emission-line ratios measured on all three spatial scales to classify the sample into star-forming, active galactic nuclei (AGN), and galaxies with a mixture of star formation and nuclear activity. We find that the relative fraction of the sample classified as star-forming versus AGN is a strong function of the integrated light enclosed by the spectroscopic aperture. We supplement our observations with a large database of nebular emission-line measurements of individual H ii regions in the SINGS galaxies culled from the literature. We use these ancillary data to conduct a detailed analysis of the radial abundance gradients and average H ii- region abundances of a large fraction of the sample. We combine these results with our new integrated spectra to estimate the central and characteristic (globally-averaged) gas-phase oxygen abundances of all 75 SINGS galaxies. We conclude with an in-depth discussion of the absolute uncertainty in the nebular oxygen abundance scale.