Our understanding of the formation and evolution of planetary systems is still fragmentary because most of the current data provide limited information about the orbital structure and dynamics of these systems. The knowledge of the orbital properties for a variety of systems and at different ages yields information on planet migration and on star-planet tidal interaction mechanisms. In this context, a long-term, multi-purpose, observational programme has started with HARPS-N at TNG and aims to characterise the global architectural properties of exoplanetary systems. The goal of this first paper is to fully characterise the orbital properties of the transiting system Qatar-1 as well as the physical properties of the star and the planet. We find that the system is well aligned and fits well within the general lambda versus Tsubeff trend. We can definitely rule out any significant orbital eccentricity. The evolutionary status of the system is inferred based on gyrochronology, and the present orbital configuration and timescale for orbital decay are discussed in terms of star-planet tidal interactions.