The Taylor–Couette flow between coaxial cylinders with a rotating inner cylinder is a key tool for achieving process intensification. However, this flow system exhibits multistability. For example, the number of Taylor cells formed stochastically depends on the hysteresis and start-up operations in the steady state. To address this issue from a practical perspective, the development process from the start of inner cylinder rotation was investigated using numerical simulations. Two types of fluids, Newtonian and shear-thinning fluids were used. First, it was confirmed that the number of Taylor cells obtained by the simulation agreed with that having the highest probability in the experiments. In addition, the cell size with shear thinning was larger than that of the Newtonian fluid. This is primarily due to the merging of the distorted cell, which is transiently formed, adjoining the end cell. Furthermore, the number of Taylor cells in a steady state with various angular accelerations was tested. Consequently, it was inferred that the number of cells is affected not only by the steady-state time, but also by the function type of the angular acceleration with time.