The formation of galactic discs and the efficiency of star formation within them are issues central to our understanding of galaxy formation. We have developed a detailed and versatile model of disc formation which combines the strengths of previous studies of isolated discs with those of hierarchical galaxy formation models. Disc structure is inferred from the distribution of angular momentum in hot halo gas and the hierarchical build-up of dark matter, leading to theoretically generated systems where the evolution of surface density, rotation, velocity dispersion, stability and metallicity is predicted for annular regions of width 20-100pc. The model will be used to establish whether the accepted theory of large-scale structure formation in the universe is consistent with observed trends in the properties of disc galaxies. This first paper explicitly examines the importance of embedding such calculations within a merging hierarchy of dark matter haloes, finding that this leads to dramatically different formation histories compared to models in which discs grow in isolation. Different models of star formation are explored, and are found to have only a secondary influence on the properties of the resulting galaxy discs, the main governing factor being the infalling gas supply from the hot halo.