We present a theoretical model for the evolution of mass, angular momentum and size of galaxy discs and bulges, and we implement it into the semi-analytic galaxy formation code, Semi-Analytic Galaxy Evolution. The model follows both secular and violent evolutionary channels, including smooth accretion, disc instabilities, minor and major mergers. We find that the combination of our recipe with hierarchical clustering produces two distinct populations of bulges: merger-driven bulges, akin to classical bulges and ellipticals, and instability-driven bulges, akin to secular (or pseudo-)bulges. The model mostly reproduces the mass–size relation of gaseous and stellar discs, the evolution of the mass–size relation of ellipticals, the Faber–Jackson relation, and the magnitude–colour diagram of classical and secular bulges. The model predicts only a small overlap of merger-driven and instability-driven components in the same galaxy, and predicts different bulge types as a function of galaxy mass and disc fraction. Bulge type also affects the star formation rate and colour at a given luminosity. The model predicts a population of merger-driven red ellipticals that dominate both the low-mass and high-mass ends of the galaxy population, and span all dynamical ages; merger-driven bulges in disc galaxies are dynamically old and do not interfere with subsequent evolution of the star-forming component. Instability-driven bulges dominate the population at intermediate galaxy masses, especially thriving in massive discs. The model green valley is exclusively populated by instability-driven bulge hosts. Through the present implementation, the mass accretion history is perceivable in the galaxy structure, morphology and colours.