A theoretical investigation has been carried out for understanding the nonlinear propagation characteristics of the dust-ion-acoustic (DIA) solitary and shock waves in an unmagnetized dusty plasma containing Maxwellian negative ions, superthermally distributed electrons, positively charged stationary dust particles and cold mobile inertial positive ions. The reductive perturbation method has been employed to derive the Korteweg-de Vries (K-dV), modified Korteweg-de Vries (mK-dV) as well as Burger equations along with its solutions in order to study the basic features of such waves both numerically and analytically. It has been found that the presence of superthermal electrons significantly modifies the basic properties of such DIA solitary and shock wave structures. The present investigation can be very effective for understanding and studying various astrophysical environments (viz. Saturn magnetosphere, pulsar magnetosphere, etc).