Robotic systems driven by artificial muscles present unique challenges due to the nonlinear dynamics of actuators and the complex designs of mechanical structures. Traditional model-based controllers often struggle to achieve desired control performance in such systems. Deep reinforcement learning (DRL), a trending machine learning technique widely adopted in robot control, offers a promising alternative. However, integrating DRL into these robotic systems faces significant challenges, including the requirement for large amounts of training data and the inevitable sim-to-real gap when deployed to real-world robots. This paper proposes an efficient reinforcement learning control framework with sim-to-real transfer to address these challenges. Bootstrap and augmentation enhancements are designed to improve the data efficiency of baseline DRL algorithms, while a sim-to-real transfer technique, namely randomization of muscle dynamics, is adopted to bridge the gap between simulation and real-world deployment. Extensive experiments and ablation studies are conducted utilizing two string-type artificial muscle-driven robotic systems including a two degree-of-freedom robotic eye and a parallel robotic wrist, the results of which demonstrate the effectiveness of the proposed learning control strategy.