This series of papers is intended to present astrocladistics in some detail and evaluate this methodology in reconstructing phylogenies of galaxies. Being based on the evolution of all the characters describing galaxies, it is an objective way of understanding galaxy diversity through evolutionary relationships. In this first paper, we present the basic steps of a cladistic analysis and show both theoretically and practically that it can be applied to galaxies. For illustration, we use a sample of 50 simulated galaxies taken from the GALICS database, which are described by 91 observables (dynamics, masses and luminosities). These 50 simulated galaxies are indeed 10 different galaxies taken at 5 cosmological epochs, and they are free of merger events. The astrocladistic analysis easily reconstructs the true chronology of evolution relationships within this sample. It also demonstrates that burst characters are not relevant for galaxy evolution as a whole. A companion paper is devoted to the formalization of the concepts of formation and diversification in galaxy evolution.