Climbing soft robots are of tremendous interest in both science and engineering due to their potential applications in intelligent surveillance, inspection, maintenance, and detection under environments away from the ground. The challenge lies in the design of a fast, robust, switchable adhesion actuator to easily attach and detach the vertical surfaces. Here, we propose a new design of pneumatic-actuated bioinspired soft adhesion actuator working both on ground and under water. It is composed of extremely soft bilayer structures with an embedded spiral pneumatic channel resting on top of a base layer with a cavity. Rather than the traditional way of directly pumping air out of the cavity for suction in hard polymer-based adhesion actuator, we inflate air into the top spiral channel to deform into a stable 3D domed shape for achieving negative pressure in the cavity. The characterization of the maximum shear adhesion force of the proposed soft adhesion actuator shows strong and rapid reversible adhesion on multiple types of smooth and semi-smooth surfaces. Based on the switchable adhesion actuator, we design and fabricate a novel load-carrying amphibious climbing soft robot (ACSR) by combining with a soft bending actuator. We demonstrate that it can operate on a wide range of foreign horizontal and vertical surfaces including dry, wet, slippery, smooth, and semi-smooth ones on ground and also under water with certain load-carrying capability. We show that the vertical climbing speed can reach about 286 mm/min (1.6 body length/min) while carrying over 200g object (over 5 times the weight of ACSR itself) during climbing on ground and under water. This research could largely push the boundaries of soft robot capabilities and multifunctionality in window cleaning and underwater inspection under harsh environment.