Assessments of the cold-gas reservoir in galaxies are a cornerstone for understanding star-formation processes and the role of feedback and baryonic cycling in galaxy evolution. Here we exploit a sample of 392 galaxies (dubbed MAGMA, Metallicity and Gas for Mass Assembly), presented in a recent paper, to quantify molecular and atomic gas properties across a broad range in stellar mass, Mstar, from ∼107 − 1011 M⊙. First, we find the metallicity (Z) dependence of the conversion factor for CO luminosity to molecular H2 mass αCO to be shallower than previous estimates, with αCO ∝ (Z/Z⊙)−1.55. Second, molecular gas mass MH2 is found to be strongly correlated with Mstar and star-formation rate (SFR), enabling predictions of MH2 good to within ∼0.2 dex; analogous relations for atomic gas mass MHI and total gas mass Mgas are less accurate, ∼0.4 dex and ∼0.3 dex, respectively. Indeed, the behavior of atomic gas mass MHI in MAGMA scaling relations suggests that it may be a third, independent variable that encapsulates information about the circumgalactic environment and gas accretion. If Mgas is considered to depend on MHI, together with Mstar and SFR, we obtain a relation that predicts Mgas to within ∼0.05 dex. Finally, the analysis of depletion times and the scaling of MHI/Mstar and MH2/Mstar over three different mass bins suggests that the partition of gas and the regulation of star formation through gas content depends on the mass regime. Dwarf galaxies (Mstar ≲ 3 × 109 M⊙) tend to be overwhelmed by (H I) accretion, and despite short τH2 (and thus presumably high star-formation efficiency), star formation is unable to keep up with the gas supply. For galaxies in the intermediate Mstar “gas-equilibrium” bin (3 × 109 M⊙ ≲ Mstar ≲3 × 1010 M⊙), star formation proceeds apace with gas availability, and H I and H2 are both proportional to SFR. In the most massive “gas-poor, bimodality” regime (Mstar ≳ 3 × 1010 M⊙), H I does not apparently participate in star formation, although it generally dominates in mass over H2. Our results confirm that atomic gas plays a key role in baryonic cycling, and is a fundamental ingredient for current and future star formation, especially in dwarf galaxies.