The Gatenby-Gawlinski model for the description of cancer invasion is taken into account with the aim of investigating its structural aspects. From a phenomenological point of view, that model is inspired by the War-burg effect and relies on the acid-mediated invasion hypothesis. By means of both an analytical and numerical approach, on the heels of the results available in the literature, the main purpose consists in paving the way for improving the understanding of the mechanisms the model is ruled by. We adopt a suitable numerical strategy based on a finite volume approximation and provide a space-averaged propagation speed estimate as a tool to investigate the phenomenon of traveling fronts. Simulations are performed and realistic evidence emerging from the experiments is highlighted. Afterwards, we make reasonable assumptions for justifying some system reductions leading to a simplified version and discuss the results without neglecting natural bonds with the whole system. Finally, the multidimensional environment is explored with emphasis on some important qualitative aspects strongly linked to experimental observations.