Recent theory has demonstrated that Kagome photonic crystals (PCs) support first-order and second-order topological phenomena. Here, we extend the topological physics of the Kagome lattice to surface electromagnetic waves and experimentally show a Kagome surface-wave PC. Under the protection of first-order and second-order topologies, both robust edge modes and in-gap corner modes are observed. The robust transport of edge modes is demonstrated by high transmission through the waveguide with a sharp bend. The localized corner mode is found at the corner with one isolated rod when a triangle-shaped sample is constructed. Our work not only shows a platform to mimic the topological physics in classical wave systems, but also offers a potential application in designing high-performance photonic devices.