The effects of relativistic dynamics and thermodynamics in the development of Kelvin-Helmholtz instabilities in planar, relativistic jets along the early phases (namely linear and saturation phases) of evolution has been studied by a combination of linear stability analysis and high-resolution numerical simulations for the most unstable first reflection modes in the temporal approach. Three different values of the jet Lorentz factor (5, 10 and 20) and a few different values of specific internal energy of the jet matter (from 0.08 to 60.0c^2) have been considered. Figures illustrating the evolution of the perturbations are also shown. Our simulations reproduce the linear regime of evolution of the excited eigenmodes of the different models with a high accuracy. In all the cases the longitudinal velocity perturbation is the first quantity that departs from the linear growth when it reaches a value close to the speed of light in the jet reference frame. The saturation phase extends from the end of the linear phase up to the saturation of the transversal velocity perturbation (at approximately 0.5c in the jet reference frame). The saturation times for the different numerical models are explained from elementary considerations, i.e. from properties of linear modes provided by the linear stability analysis and from the limitation of the transversal perturbation velocity. The limitation of the components of the velocity perturbation at the end of the linear and saturation phases allows us to conclude that the relativistic nature of the flow appears to be responsible for the departure of the system from linear evolution. The high accuracy of our simulations in describing the early stages of evolution of the KH instability (as derived from the agreement between the computed and expected linear growth rates and the consistency of the saturation times) establishes a solid basis to study the fully nonlinear regime, to be done elsewhere. The present paper also sets the theoretical and numerical background for these further studies.