In this paper, a (3+1)-dimensional nonlinear evolution equation is investigated, which can be used to describe reacting mixtures and shallow water waves. Through the Hirota method and symbolic computation, bilinear forms and Backlund transformation are derived, which are different from those in the existing literature. Moreover, N-shock-wave solutions are obtained. Based on those shock-wave solutions, propagation and collision of the shock waves are discussed via the asymptotic and graphic analysis on different planes: (1) oblique elastic collisions between/among the two/three shock waves will arise on the x-y and y-z planes, while parallel elastic collisions exist on the x-z plane; (2) shock waves maintain their original directions, amplitudes and velocities except for some small phase shifts after each collision; (3) the shock wave with higher amplitude travels faster and moves across the slower.