The possibility of attaining chiral edge modes under periodic driving has spurred tremendous attention both theoretically and experimentally, especially in light of anomalous Floquet topological phases that feature vanishing Chern numbers unlike any static counterpart. We consider here a periodically modulated honeycomb lattice and experimentally relevant driving protocols, which allows us to obtain edge modes of various character in a simple model. We calculate the phase diagram over a wide range of parameters and recover an anomalous topological phase with quasienergy gaps harboring edge states with opposite chirality. Motivated by the advances in single-site control in optical lattices, we investigate wave-packet dynamics localized at the edges in distinct Floquet topological regimes that cannot be achieved in equilibrium. We analyze transport properties in edge modes which originate from the same bands but with support at different quasienergies and sublattices as well as possessing different chiralities. We find that an anomalous Floquet topological phase can in general generate more robust chiral edge motion than a Haldane phase, allowing for more effective loading of the wave packet into edge channels. Our results demonstrate that the rich interplay of wave-packet dynamics and topological edge states can serve as a versatile tool in ultracold quantum gases in optical lattices.