The Cassini mission unveiled the intense and diverse activity in Saturn's atmosphere: banded jets, waves, vortices, equatorial oscillations. To set the path towards a better understanding of those phenomena, we performed high-resolution multi-annual numerical simulations of Saturn's atmospheric dynamics. We built a new Global Climate Model [GCM] for Saturn, named the Saturn DYNAMICO GCM, by combining a radiative-seasonal model tailored for Saturn to a hydrodynamical solver based on an icosahedral grid suitable for massively-parallel architectures. The impact of numerical dissipation, and the conservation of angular momentum, are examined in the model before a reference simulation employing the Saturn DYNAMICO GCM with a 1/2° latitude-longitude resolution is considered for analysis. Mid-latitude banded jets showing similarity with observations are reproduced by our model. Those jets are accelerated and maintained by eddy momentum transfers to the mean flow, with the magnitude of momentum fluxes compliant with the observed values. The eddy activity is not regularly distributed with time, but appears as bursts; both barotropic and baroclinic instabilities could play a role in the eddy activity. The steady-state latitude of occurrence of jets is controlled by poleward migration during the spin-up of our model. At the equator, a weakly-superrotating tropospheric jet and vertically-stacked alternating stratospheric jets are obtained in our GCM simulations. The model produces Yanai (Rossby-gravity), Rossby and Kelvin waves at the equator, as well as extratropical Rossby waves, and large-scale vortices in polar regions. Challenges remain to reproduce Saturn's powerful superrotating jet and hexagon-shaped circumpolar jet in the troposphere, and downward-propagating equatorial oscillation in the stratosphere.