We present the transmission spectrum of HAT-P-12b through a joint analysis of data obtained from the Hubble Space Telescope Space Telescope Imaging Spectrograph and Wide Field Camera 3 and Spitzer, covering the wavelength range 0.3-5.0 μm. We detect a muted water vapor absorption feature at 1.4 μm attenuated by clouds, as well as a Rayleigh scattering slope in the optical indicative of small particles. We interpret the transmission spectrum using both the state-of-the-art atmospheric retrieval code SCARLET and the aerosol microphysics model CARMA. These models indicate that the atmosphere of HAT-P-12b is consistent with a broad range of metallicities between several tens to a few hundred times solar, a roughly solar C/O ratio, and moderately efficient vertical mixing. Cloud models that include condensate clouds do not readily generate the submicron particles necessary to reproduce the observed Rayleigh scattering slope, while models that incorporate photochemical hazes composed of soot or tholins are able to match the full transmission spectrum. From a complementary analysis of secondary eclipses by Spitzer, we obtain measured depths of 0.042% ± 0.013% and 0.045% ± 0.018% at 3.6 and 4.5 μm, respectively, which are consistent with a blackbody temperature of 890+60-70 K and indicate efficient day-night heat recirculation. HAT-P-12b joins the growing number of well-characterized warm planets that underscore the importance of clouds and hazes in our understanding of exoplanet atmospheres.