Linear and nonlinear dust-acoustic wave propagation in a non-ideal classical dusty plasma consisting of electrons, ions and dust grains is investigated by incorporating the van der Waals equation of state for the dust component. For linear waves, it is found from the normal mode dispersion relation that the volume reduction coefficient enhances the phase speed of the dust-acoustic waves, while the molecular cohesive forces lead to a decrease in the phase speed. The relative magnitudes of the two contributions depend on the specific parameter regimes characterizing the non-ideal nature of the dust component. In the high-temperature limit, there is a net increase in the dust-acoustic phase speed, while near the critical point the phase speed is reduced when compared with that for the ideal-gas case. For large amplitudes, we discuss the existence of dust-acoustic solitons by deriving the exact Sagdeev potential. While supersonic solitons are found to be admissible in both the sub- and supercritical parameter regimes, subsonic propagation near the dust-acoustic speed is possible only for the supercritical case. For small but finite amplitudes, explicit analytical solutions have been obtained. A limiting case of these solutions is also discussed.