Using the adaptive time-dependent density matrix renormalization group, we study the time evolution of strongly correlated bosonic and fermionic systems on one-dimensional lattices pushed out of equilibrium. In particular, we investigate a system of interacting spinless fermions after a sudden change in the interaction strength and a system of soft- core bosons modeled by the Bose-Hubbard model released from a trapping box-potential. By considering the momentum distribution function of the bosonic system, we demonstrate the formation of (quasi-)coherent matter waves emerging from an initial Mott insulating state over a wide range of the interaction strength between the particles. In the fermionic system, we find the density correlations to exhibit a characteristic light-cone-like time evolution which is representative of a ballistic transport of information. Such behavior is observed both when quenching an insulator into the metallic region and also when quenching within the insulating region, but not when a metallic state is quenched deep into the insulating regime. Instead, stable domain walls in the density correlations emerge during the time evolution, consistent with the predictions of the Kibble-Zurek mechanism.