To break the application scenario limitations of traditional bionic underwater robots and open up the way of information docking between underwater and water surface, a systematic study was conducted on the cross-medium standing-and-walking (SAW) behavior of dolphins under the collaborative movements of the body, caudal, and pectoral fins. A three-dimensional physical model of the biomimetic dolphin robot was established, a collaborative movement law of the body, caudal, and pectoral fins was proposed, and the SAW behavior under two modes, Homologous and Reverse modes, was realized. The mapping relationship between the robot's kinematic parameters and hydrodynamic performance parameters was analyzed, the respective advantages of the two vertical walking modes were compared, and the SAW mechanism of the two modes was revealed physically with the help of the evolution law of the flow field around the robot. The results show that the biomimetic dolphin robot can realize cross-medium SAW behavior through the collaborative movements of the body, caudal pectoral fins. It is worth noting that the Hom mode has a superior walking speed, which can reach up to 0.27 m/s, an increase in 83.3% compared to the Rev mode under the same conditions, while the Rev mode has better walking stability, with a minimum fluctuation rate of 2.59%, a 30.8% improvement compared to the Hom mode. This research provides a novel idea for enhancing the surface operation capability of traditional biomimetic underwater robots and provides an important fluid mechanics theoretical basis for the design of new generation of cross-medium dolphin robots.