Monitoring, sensing, and exploration of over 70% of the Earth's surface that is covered with water is permitted through the deployment of underwater bioinspired robots without affecting the natural habitat. To create a soft robot actuated with soft polymeric actuators, this paper describes the development of a lightweight jellyfish-inspired swimming robot, which achieves a maximum vertical swimming speed of 7.3 mm/s (0.05 body length/s) and is characterized by a simple design. The robot, named Jelly-Z, utilizes a contraction-expansion mechanism for swimming similar to the motion of a Moon jellyfish. The objective of this paper is to understand the behavior of soft silicone structure actuated by novel self-coiled polymer muscles in an underwater environment by varying stimuli and investigate the associated vortex for swimming like a jellyfish. To better understand the characteristics of this motion, simplified Fluid-structure simulation, and particle image velocimetry (PIV) tests were conducted to study the wake structure from the robot's bell margin. The thrust generated by the robot was also characterized with a force sensor to ascertain the force and cost of transport (COT) at different input currents. Jelly-Z is the first robot that utilized twisted and coiled polymer fishing line (TCP) actuators for articulation of the bell and showed successful swimming operations. Here, a thorough investigation on swimming characteristics in an underwater setting is presented theoretically and experimentally. We found swimming metrics of the robot are comparable with other jellyfish-inspired robots that have utilized different actuation mechanisms, but the actuators used here are scalable and can be made in-house relatively easily, hence paving way for further advancements into the use of these actuators.