As tire forces approach road friction limits, vehicles may always exhibit performance degradation and even instability. The actual capacity of the available road friction imposes coupled limits on a vehicle’s longitudinal and lateral accelerations. In this paper, a varying speed method is proposed to design feasible speeds and accelerations, which ensure that the autonomous vehicle will not reach the tire-road friction limits during traversing a clothoid-based path. With the consideration of uncertain traction losses and vehicle parameters, a second-order super-twisting (ST) based speed control strategy is proposed to track above feasible speeds based on varying speed method, and to stabilize the sliding-mode variable of the speed tracking error with relative degree 1. Meanwhile, a second-order quasi-continuous (QC) based path-following control strategy is proposed to follow a desired transition path, and to stabilize the sliding-mode variable of the composite path-following errors with relative degree 2. On this basis, the proposed controllers have been verified to lead good robustness for tracking the ideal speeds and following the desired paths. As compared with the boundary of the autonomous vehicle running at a constant speed, the feasible speed boundary using varying speed method without exceeding the tire-road friction limits can be enlarged up to about 1.59 times, which is decided by the direction change between the entry and exit points of the desired path.