Tuning and stabilising topological states, such as Weyl semimetals, Dirac semimetals, or topological insulators, is emerging as one of the major topics in materials science. Periodic driving of many-body systems offers a platform to design Floquet states of matter with tunable electronic properties on ultrafast time scales. Here we show by first principles calculations how femtosecond laser pulses with circularly polarised light can be used to switch between Weyl semimetal, Dirac semimetal, and topological insulator states in a prototypical 3D Dirac material, Na$_3$Bi. Our findings are general and apply to any 3D Dirac semimetal. We establish the concept of time-dependent bands and steering of Floquet-Weyl points (Floquet-WPs), and demonstrate how light can enhance topological protection against lattice perturbations. Our work has potential practical implications for the ultrafast switching of materials properties, like optical band gaps or anomalous magnetoresistance. Moreover, we introduce Floquet time-dependent density functional theory (Floquet-TDDFT) as a general and robust first principles method for predictive Floquet engineering of topological states of matter.