The nonholonomic wheeled mobile robot is widely used in scenarios such as military equipment, transportation, medical, and household services. In this paper, a smooth feedback control strategy is used to solve the target point stabilization and trajectory tracking problems of such robots, so that the mobile robot can adaptively and smoothly switch between stabilization and tracking control. Firstly, a unified mathematical model is constructed for the stabilization and tracking problems of nonholonomic wheeled mobile robots. Then, combined with the theory of uniform delta persistency of excitation, a novel controller has been designed based on time-varying functions. This controller can not only obtain smoother autonomous motion trajectory and faster convergence rate without prior knowledge of reference trajectory when addressing the point stabilization problem, but also complete the tracking of the desired trajectory, and achieve fast tracking in the case of the desired trajectory switching. In addition, we give the stability analysis of stabilization and tracking error using the Lyapunov stability theory. Finally, the feasibility and effectiveness of the proposed method are verified by numerical simulations and experiments.