This series of papers aims at understanding the formation and evolution of non-barred disc galaxies. We use the new spectro-photometric decomposition code, C2D, to separate the spectral information of bulges and discs of a statistically representative sample of galaxies from the CALIFA survey. Then, we study their stellar population properties analysing the structure-independent datacubes with the PIPE3D algorithm. We find a correlation between the bulge-to-total (B/T) luminosity (and mass) ratio and galaxy stellar mass. The B/T mass ratio has only a mild evolution with redshift, but the bulge-to-disc (B/D) mass ratio shows a clear increase of the disc component since redshift z < 1 for massive galaxies. The mass-size relation for both bulges and discs describes an upturn at high galaxy stellar masses (log (M⋆/M⊙) > 10.5). The relation holds for bulges but not for discs when using their individual stellar masses. We find a negligible evolution of the mass-size relation for both the most massive (log(M⋆,b,d/M⊙) > 10) bulges and discs. For lower masses, discs show a larger variation than bulges. We also find a correlation between the Sérsic index of bulges and both galaxy and bulge stellar mass, which does not hold for the disc mass. Our results support an inside-out formation of nearby non-barred galaxies, and they suggest that (i) bulges formed early-on and (ii) they have not evolved much through cosmic time. However, we find that the early properties of bulges drive the future evolution of the galaxy as a whole, and particularly the properties of the discs that eventually form around them.