Aims: We present a new semi-analytical model of galaxy formation, GECO (Galaxy Evolution COde), designed to improve our understanding of when and how the processes of both star formation and galaxy assembly took place, by comparison with a wide variety of data about galaxy mass-function evolution and star-formation histories. Methods: Our model is structured into a Monte Carlo algorithm based on the extended Press-Schechter theory, to represent the merging hierarchy of dark matter halos, and a set of analytic algorithms to treat the baryonic physics, including classical recipes for gas cooling, star-formation timescales, galaxy mergers, and supernova (SN) feedback. In addition to the galaxies, the parallel growth of BHs is followed in time, and their feedback on the hosting galaxies is modelled. We set the model free parameters by matching data on local stellar mass functions and the relation between galaxy bulge and black-hole mass at z = 0. Results: Based on these local boundary conditions, we investigate how data on the high-redshift universe constrain our understanding of the physical processes driving the evolution, focusing in particular on the assembly of stellar mass and the star-formation history of galaxies. Since both processes are currently strongly constrained by cosmological near- and far-IR surveys with the Spitzer Space Telescope, the basic physics of the Λ CDM hierarchical clustering concept of galaxy formation can be effectively tested by us by comparison with the most reliable set of observables using a minimal number of free parameters. Conclusions: Our investigation shows that when the timescales of star formation and mass assembly are studied as a function of dark matter halo mass and a given galaxy stellar mass, the “downsizing” fashion of star formation appears to be a natural outcome of the model, being reproduced even in the absence of the AGN feedback. In contrast, the stellar mass assembly history turns out to follow a more standard hierarchical pattern that is progressive with cosmic time, the more massive systems being assembled at later times mainly through dissipationless mergers.