We study transient dynamics in a two-dimensional system of interacting Dirac fermions subject to a quenched drive with circularly polarized light. In the absence of interactions, the drive opens a gap at the Dirac point in the quasienergy spectrum, inducing nontrivial band topology. We investigate the dynamics of this gap opening process, taking into account the essential role of electron-electron interactions. Crucially, scattering due to interactions (1) induces dephasing, which erases memory of the system's prequench state and yields the intrinsic timescale for gap emergence, and (2) provides a mechanism for the system to absorb energy of the drive, leading to heating which must be mitigated to ensure the success of Floquet band engineering. We characterize the gap opening process via the system's generalized spectral function and correlators probed by photoemission experiments, and we identify a parameter regime at moderate driving frequencies where a hierarchy of timescales allows a well-defined Floquet gap to be produced and studied before the deleterious effects of heating set in.