We study the stochastic dynamics of a particle with two distinct motility states. Each oneis characterized by two parameters: one represents the average speed and the otherrepresents the persistence quantifying the tendency to maintain the current direction ofmotion. We consider a run-and-tumble process, which is a combination of an activefast motility mode (persistent motion) and a passive slow mode (diffusion). Assumingstochastic transitions between the two motility states, wederive an analytical expressionfor the time evolution of the mean square displacement. The interplay of the keyparameters and the initial conditions as for instance the probability of initially starting inthe run or tumble state leads to a variety of transient regimes of anomalous transport ondifferent time scales before approaching the asymptotic diffusive dynamics. We estimatethe crossover time to the long-term diffusive regime and prove that the asymptoticdiffusion constant is independent of initially starting inthe run or tumble state.