A path following feedback controller for mobile robots based on model predictive control (MPC) is introduced. The path following problem is formulated in the Frenet-Serret frame with the predictive controller aiming to drive the mobile robot approach and follow a parametrized geometric path while maximizing the robot speed, i.e., the covered robot distance. Since hyperparameters of the MPC formulation greatly affect the tradeoff between path-tracking error and maximum possible speed, we propose a Bayesian optimization-based algorithm to automatically select suitable parameters that maximize the speed while keeping the tracking error low. The algorithms are implemented on an omnidirectional mobile robot and validated using different predefined geometric paths. Finally, we compare the performance of the robot with manual tuning and automatic tuning, thereby illustrating the effectiveness of the latter.