  We implement physically motivated recipes for partitioning cold gas into
different phases (atomic, molecular, and ionized) in galaxies within
semi-analytic models of galaxy formation based on cosmological merger trees. We
then model the conversion of molecular gas into stars using empirical recipes
motivated by recent observations. We explore the impact of these new recipes on
the evolution of fundamental galaxy properties such as stellar mass, star
formation rate (SFR), and gas and stellar phase metallicity. We present
predictions for stellar mass functions, stellar mass vs. SFR relations, and
cold gas phase and stellar mass-metallicity relations for our fiducial models,
from redshift $z\sim 6$ to the present day. In addition we present predictions
for the global SFR, mass assembly history, and cosmic enrichment history. We
find that the predicted stellar properties of galaxies (stellar mass, SFR,
metallicity) are remarkably insensitive to the details of the recipes used for
partitioning gas into HI and H$_2$. We see significant sensitivity to the
recipes for H$_2$ formation only in very low mass halos, which host galaxies
that are not detectable with current observational facilities except very
nearby. The properties of low-mass galaxies are also quite insensitive to the
details of the recipe used for converting H$_2$ into stars, while the formation
epoch of massive galaxies does depend on this significantly. (Abridged)
