  In the Milky Way, cosmic rays (CRs) are dynamically important in the
interstellar medium, contribute to hydrostatic balance, and may help regulate
star formation. However, we know far less about the importance of CRs in
galaxies whose gas content or star formation rate differ significantly from
those of the Milky Way. Here we construct self-consistent models for hadronic
CR transport, losses, and contribution to pressure balance as a function of
galaxy properties, covering a broad range of parameters from dwarfs to extreme
starbursts. While the CR energy density increases from $\sim 1$ eV cm$^{-3}$ to
$\sim 1$ keV cm$^{-3}$ over the range from sub-Milky Way dwarfs to bright
starbursts, strong hadronic losses render CRs increasingly unimportant
dynamically as the star formation rate surface density increases. In Milky
Way-like systems, CR pressure is typically comparable to turbulent gas and
magnetic pressure at the galactic midplane, but the ratio of CR pressure to gas
pressure drops to $\sim 10^{-3}$ in dense starbursts. Galaxies also become
increasingly CR calorimetric and gamma-ray bright in this limit. The degree of
calorimetry at fixed galaxy properties is sensitive to the assumed model for CR
transport, and in particular to the time CRs spend interacting with neutral
ISM, where they undergo strong streaming losses. We also find that in some
regimes of parameter space hydrostatic equilibrium discs cannot exist, and in
Paper II of this series we use this result to derive a critical surface in the
plane of star formation surface density and gas surface density beyond which
CRs may drive large-scale galactic winds.
