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 $Deltapi_1$, pressure offset $epsilon_p$, gravity offset $epsilon_g$, and coupling factor $q$ as a function of $Delta u$ 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 $Deltapi_1$ and $Delta u$, 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 $Delta u$ 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.
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