We present spectroscopic observations of the host galaxies of 82 low-redshift Type Ia supernovae (SNe Ia) discovered by the Palomar Transient Factory. We determine star formation rates, gas-phase/stellarmetallicities, and stellar masses and ages of these objects. As expected, strong correlations between the SN Ia light-curve width (stretch) and the host age/mass/metallicity are found: fainter, faster declining events tend to be hosted by older/massive/metal-rich galaxies. There is some evidence that redder SNe Ia explode in higher metallicity galaxies, but we found no relation between the SN colour and host galaxy extinction based on the Balmer decrement, suggesting that the colour variation of these SNe does not primarily arise from this source. SNe Ia in higher mass/metallicity galaxies also appear brighter after stretch/colour corrections than their counterparts in lower mass hosts, and the stronger correlation is with gas-phase metallicity suggesting this may be the more important variable. We also compared the host stellar mass distribution to that in galaxy-targeted SN surveys and the high-redshift untargeted Supernova Legacy Survey (SNLS). SNLS has many more low-mass galaxies, while the targeted searches have fewer. This can be explained by an evolution in the galaxy stellar mass function, coupled with an SN delay-time distribution proportional to t(-1). Finally, we found no significant difference in the mass-metallicity relation of our SN Ia hosts compared to field galaxies, suggesting any metallicity effect on the SN Ia rate is small.