Transportation system is facing a sharp disruption since the Connected Autonomous Vehicles (CAVs) can free people from driving and provide good driving experience with the aid of Vehicle-to-Vehicle (V2V) communications. Although CAVs bring benefits in terms of driving safety, vehicle string stability, and road traffic throughput, most existing work aims at improving only one of these performance metrics. However, these metrics may be mutually competitive, as they share the same communication and computing resource in a road segment. From the perspective of joint optimizing driving safety, vehicle string stability, and road traffic throughput, there is a big research gap to be filled on the resource management for connected autonomous driving. In this paper, we first explore the joint optimization on driving safety, vehicle string stability, and road traffic throughput by leveraging on the consensus Alternating Directions Method of Multipliers algorithm (ADMM). However, the limited communication bandwidth and on-board processing capacity incur the resource competition in CAVs. We next analyze the multiple tasks competition in the contention based medium access to attain the upper bound delay of V2V-related application offloading. An efficient sleeping multi-armed bandit tree-based algorithm is proposed to address the resource assignment problem. A series of simulation experiments are carried out to validate the performance of the proposed algorithms.