We calculate the dynamics of one- and two-body correlation functions in a homogeneous Bose gas at zero temperature following a sudden change in the interaction strength, in the continuum and in a lattice. By choosing suitable examples, we highlight features in the correlation functions that emerge due to the interactions and the band structure. We find that interactions dramatically change the way correlations build up and subsequently decay following a quench. For example, the Bogoliubov dispersion induces a crossover from diffusive spreading of short-range correlations to ballistic spreading of longer-range correlations. In the lattice, the correlation functions develop additional features absent in the continuum. Most strikingly, the lattice induces an additional velocity scale and some features propagate with that velocity. Finally, we discuss the ultra-short-range properties of the density-density correlation function following a quench, and the implications for experiments using this quantity to probe the "contact." Our calculations, which can be readily tested in current experiments, suggest that the dynamics of correlations may be a useful tool for extracting many-body parameters. © 2013 American Physical Society.