  This study is the first of a series of papers that provide a technique to
analyse the mixed-modes frequency spectra and characterise the structure of
stars on the subgiant and red-giant branches. We define seismic indicators,
relevant of the stellar structure and study their evolution on a grid of
models. The proposed method, EGGMiMoSA, relies on the asymptotic description of
mixed modes, defines initial guesses for the parameters, and uses a
Levenberg-Marquardt technique to adjust the mixed-modes pattern efficiently. We
follow the evolution of the mixed-modes parameters along a grid of models from
the subgiant phase to the RGB bump and extend past works. We show the impact of
the mass and composition on their evolution. The evolution of the period
spacing $\Delta\pi_1$, pressure offset $\epsilon_p$, gravity offset
$\epsilon_g$, and coupling factor $q$ as a function of $\Delta\nu$ is little
affected by the chemical composition and it follows two different regimes
depending on the evolutionary stage. On the subgiant branch, the models display
a moderate core-envelope density contrast. The evolution of $\Delta \pi_1$,
$\epsilon_p$, $\epsilon_g$, and $q$ thus significantly changes with the mass.
Also, we demonstrate that, at fixed Z/X and with proper measurements of
$\Delta\pi_1$ and $\Delta\nu$, we may unambiguously constrain the mass, radius
and age of a subgiant star. Conversely, on the red-giant branch, the
core-envelope density contrast becomes very large. Consequently, the evolution
of $\epsilon_p$, $\epsilon_g$ and $q$ as a function of $\Delta\nu$ becomes
independent of the mass. This is also true for $\Delta \pi_1$ in stars with
masses $\lesssim 1.8M_\odot$ because of core electron degeneracy. This
degeneracy is lifted for higher masses, again allowing for a precise
measurement of the age. Overall, our computations qualitatively agree with past
observed and theoretical studies.
