Recently, the task offloading and resource allocation (TORA) issues in mobile edge computing system have attracted much attention. However, the impact of different movement speeds on TORA problem and customized services for different service requirements have been largely ignored. These may severely influence the user’s quality of service (QoS). Accordingly, a speed-aware and customized TORA scheme is proposed to minimize total service latency (consisted of execution delay and migration cost) for mobile users. First, the Advantage Actor Critic (A2C) algorithm is applied to dynamically decide on which computation node to execute the tasks. Second, we assign the user’s priorities based on movement speeds and tolerable delay to allocate the wireless and computation resources respectively. Finally, the migration cost is introduced into objective model for further weakening the impact of service migration on user’s QoS. Simulation results demonstrate the superiority and effectiveness of the proposed algorithm compared with the baselines.