As robots begin to move from structured industrial environments to the real world, they must be equipped to not only safely interact with the environment, but also reason about how to leverage contact to perform tasks. In this work, we develop a modeling and motion planning framework for continuum robots that accounts for contact anywhere along the robot. We first present an analytical model for continuum manipulators under contact and discuss the ideal choice of generalized coordinates given properties of the manipulator and task specifications. We then demonstrate the utility of our model by developing a motion planning framework that can solve a diverse set of tasks. We apply our framework to end effector path planning for a soft arm in an obstacle-rich environment, and grasp planning for soft robotic grippers, where contact can happen anywhere on the arm or gripper. Finally, we verify the utility of our model and planning framework by planning a grasp with a desired contact force for a soft antipodal gripper and testing this grasp in a hardware demonstration. Overall, our model and planning approach further enhance soft and continuum robots where they already excel: utilizing contact with the world to achieve their goals with a gentle touch.