Photoevaporation of planet-forming discs by high-energy radiation from the central star is potentially a crucial mechanism for disc evolution and it may play an important role in the formation and evolution of planetary systems. We present here a new generation of X-ray photoevaporation models for solar-type stars, based on hydrodynamical simulations, which account for stellar irradiation via a significantly improved parametrization of gas temperatures, based on detailed photoionization and radiation transfer calculations. This is the first of a series of papers aiming at providing a library of models which cover the observed parameter space in stellar and disc mass, metallicity, and stellar X-ray properties. We focus here on solar-type stars (0.7 M_⊙) with relatively low-mass discs (1 per cent of the stellar mass) and explore the dependence of the wind mass-loss rates on stellar X-ray luminosity. We model primordial discs and transition discs at various stages of evolution. Our two-dimensional hydrodynamical models are then used to derive simple recipes for the mass-loss rates that are suitable for one-dimensional disc evolution and/or planet formation models typically employed for population synthesis studies. Line profiles from typical wind diagnostics ([O i]6300 Å and [Ne ii]12.8 m) are also calculated for our models and found to be roughly in agreement with previous studies. Finally, we perform a population study of transition discs by means of one-dimensional viscous evolution models including our new photoevaporation prescription and find that roughly a half of observed transition discs cavities and accretion rates could be reproduced by our models.