We use a sample of 1338 spectroscopically confirmed and photometrically classified Type Ia supernovae (SNe Ia) sourced from Carnegie Supernova Project, Center for Astrophysics Supernova Survey, Sloan Digital Sky Survey-II, and SuperNova Legacy Survey SN samples to examine the relationships between SNe Ia and the galaxies that host them. Our results provide confirmation with improved statistical significance that SNe Ia, after standardization, are on average more luminous in massive hosts (significance >5{sigma}), and decline more rapidly in massive hosts (significance >9{sigma}) and in hosts with low specific star formation rates (significance >8{sigma}). We study the variation of these relationships with redshift and detect no evolution. We split SNe Ia into pairs of subsets that are based on the properties of the hosts and fit cosmological models to each subset. Including both systematic and statistical uncertainties, we do not find any significant shift in the best-fit cosmological parameters between the subsets. Among different SN Ia subsets, we find that SNe Ia in hosts with high specific star formation rates have the least intrinsic scatter ({sigma}_int_=0.08+/-0.01) in luminosity after standardization.