This is the first paper of a series that describes the methods and basic results of the GalICS model (for Galaxies In Cosmological Simulations). GalICS is a hybrid model for hierarchical galaxy formation studies, combining the outputs of large cosmological N-body simulations with simple, semi-analytic recipes to describe the fate of the baryons within dark matter halos. The simulations produce a detailed merging tree for the dark matter halos including complete knowledge of the statistical properties arising from the gravitational forces. We intend to predict the overall statistical properties of galaxies, with special emphasis on the panchromatic spectral energy distribution emitted by galaxies in the UV/optical and IR/submm wavelength ranges. In this paper, we outline the physically motivated assumptions and key free parameters that go into the model, comparing and contrasting with other parallel efforts. We specifically illustrate the success of the model in comparison to several datasets, showing how it is able to predict the galaxy disc sizes, colours, luminosity functions from the ultraviolet to far infrared, the Tully--Fisher and Faber--Jackson relations, and the fundamental plane in the local universe. We also identify certain areas where the model fails, or where the assumptions needed to succeed are at odds with observations, and pay special attention to understanding the effects of the finite resolution of the simulations on the predictions made. Other papers in this series will take advantage of different data sets available in the literature to extend the study of the limitations and predictive power of GalICS, with particular emphasis put on high-redshift galaxies.