A long-term multi-purpose observational programme has started with HARPS-N@TNG aimed to characterise the global architectural properties of exoplanetary systems. In this first paper we fully characterise the transiting system Qatar-1. We exploit HARPS-N high-precision radial velocity measurements obtained during a transit to measure the Rossiter-McLaughlin effect in the Qatar-1 system, and out-of-transit measurements to redetermine the spectroscopic orbit. New photometric transit light-curves are analysed and a spectroscopic characterisation of the host star atmospheric parameters is performed based on various methods (line equivalent width ratios, spectral synthesis, spectral energy distribution). We achieved a significant improvement in the accuracy of the orbital parameters and derived the spin-orbit alignment of the system; this information, combined with the spectroscopic determination of the host star properties, allows us to derive the fundamental physical parameters for star and planet (masses and radii). The orbital solution for the Qatar-1 system is consistent with a circular orbit and the system presents a sky-projected obliquity of lambda = -8.4+-7.1 deg. The planet, with a mass of 1.33+-0.05 M_J, is found to be significantly more massive than previously reported. The host star is confirmed to be metal-rich ([Fe/H]= 0.20+-0.10) and slowly rotating (vsinI = 1.7+-0.3 km/s), though moderately active, as indicated by strong chromospheric emission in the Ca II H&K line cores (logR_HK about -4.60). The system is well aligned and fits well within the general lambda vs Teff trend. We 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.
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