A four-fold-degenerate three-dimensional (3D) Dirac point, represents a degenerate pair of Weyl points carrying opposite chiralities. Moreover, 3D Dirac crystals have shown many exotic features different from those of Weyl crystals. How these features evolve from 3D Dirac to Weyl crystals is important in research on 3D topological matter. Here, we realized a pair of 3D acoustic Dirac points from band inversion in a hexagonal sonic crystal and observed the surface states and helical interface states connecting the Dirac points. Furthermore, each Dirac point can transition into a pair of Weyl points with the introduction of chiral hopping. The exotic features of the surface states and interface states are inherited by the resulting Weyl crystal. Our work may serve as an ideal platform for exploring exotic physical phenomena in 3D topological semimetals.