We design a topological phononic crystal in a thin plate, aiming at the development of an efficient elastic waveguide based on the concept of band topology in phonon dispersion. We adopt a snowflake-like structure for the crystal unit cell and search for the optimal structure that exhibits the topological phase transition of the three-dimensional phononic crystal by changing the unit cell structure. The bandgap width can be adjusted by varying the length of the branch of the snow side, and a topological phase transition occurs at the unit cell structure with three-fold rotational symmetry. Elastic waveguides based on edge modes appearing at interfaces between crystals with different band topologies are designed, and their transmission efficiency is evaluated numerically and experimentally. The results demonstrate robustness of the elastic wave propagation in thin plates. Moreover, we experimentally estimate a robustness of the topologically protected propagating states against structural inhomogeneities. The results obtained in this study pave the way for practical development of new surface acoustic wave technologies for high-frequency communication devices.