The Differential Transform Method (DTM) is an analytical and numerical method for solving a wide variety of differential equations and usually gets the solution in a series form. The multi-step DTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions. In this paper, this new algorithm is applied to a class of density dependent diffusion equations with memory-delay effect. The multistep differential transform solutions for various strengths of the density dependence along with bounds on the range of the convergence are obtained. The numerical solutions are obtained by the Runge-Kutta-Fehlberg 45 method. Then, a comparative study between the multi-step DTM and Runge-Kutta-Fehlberg 45 method is presented. The results demonstrate reliability and efficiency of the algorithm developed. Finally, the dependence of the traveling wave solutions on various parameters, particularly the memory-delay term, is discussed.