Although the spectral lines of hydrogen contain valuable information on the physical properties of a variety of astrophysical plasmas, including the upper solar chromosphere, relatively little is known about their scattering polarization signals, whose modification via the Hanle effect may be exploited for magnetic field diagnostics. Here we report on a basic theoretical investigation of the linear polarization produced by scattering processes and the Hanle effect in Lyα, Lyβ, and Hα taking into account multilevel radiative transfer effects in an isothermal stellar atmosphere model, the fine-structure of the hydrogen levels, as well as the impact of collisions with electrons and protons. The main aim of this first paper is to elucidate the key physical mechanisms that control the emergent fractional linear polarization in the three lines, as well as its sensitivity to the perturbers' density and to the strength and structure of microstructured and deterministic magnetic fields. To this end, we apply an efficient radiative transfer code we have developed for performing numerical simulations of the Hanle effect in multilevel systems with overlapping line transitions. For low-density plasmas, such as that of the upper solar chromosphere, collisional depolarization is caused mainly by collisional transitions between the fine-structure levels of the n = 3 level, so that it is virtually insignificant for Lyα but important for Lyβ and Hα. We show the impact of the Hanle effect on the three lines taking into account the radiative transfer coupling between the different hydrogen line transitions. For example, we demonstrate that the linear polarization profile of the Hα line is sensitive to the presence of magnetic field gradients in the line core formation region, and that in solar-like chromospheres selective absorption of polarization components does not play any significant role in the emergent scattering polarization.