This paper presents a three-dimensional (3-D) path planning method that combines the gliding with dolphin-like motions for the gliding robotic dolphin. A specific task that the robot uses the gliding motion for long-distance cruise and the dolphin-like motion for maneuverable obstacle avoidance is employed. The results of simulations and aquatic experiments validate the full-state dynamic model and the specific task, further offer some theoretical supports for 3-D path planning. Further, the 3-D path planning method is composed of three main components: 1) gliding path generation; 2) improved Astar (A*) algorithm; and 3) segmented Bezier curve smoothing. First, the gliding path is generated autonomously with the kinematic constraints that are obtained via the simulations of dynamic model. Furthermore, when the obstacles are detected by the sonar, an improved A* algorithm is employed to avoid the obstacles. Afterward, considering the path planned by A* is unsmoothed, a segmented Bezier curve method is presented. Simulation results demonstrate the effectiveness of the method, offering valuable insight into the utilization of hybrid underwater robots in the context of real-time task execution.