Soft robots enable safe, adaptive interaction in environments where rigid systems are inadequate, but their continuous, deformable nature makes modeling and control challenging. This work presents a data-driven framework that uses supervised learning and reinforcement learning to model the kinematics of a soft robot, develop task-driven control policies, and form safety filters. Using experimental data from a custom silicone-based, shape memory alloy (SMA)-actuated limb, we train an expressive kinematics model that captures the nonlinear, temperature-dependent behavior of SMA and silicone materials. This model is integrated into a reinforcement learning setup, allowing the robot to acquire control strategies to achieve tasks without relying on computationally expensive simulations. Furthermore, by leveraging the model as part of the learned policy, we preserve performance while reducing training time. To ensure safety, the framework uses a novel reward formulation to enable the training of a safety filter, which rejects actions that would lead the system into states that could be damaging to itself or others, thereby protecting both the robot and its environment from unsafe actions. Experimental results demonstrate robust task performance, good motion prediction, and adherence to safety constraints.