Context. The study of exoplanets at different evolutionary stages can shed light on their formation, migration, and evolution. The determination of exoplanet properties depends on the properties of their host stars. It is therefore important to characterise the host stars for accurate knowledge on their planets. Aims. Our final goal is to derive, in a homogeneous and accurate way, the stellar atmospheric parameters and elemental abundances of ten young TESS (Transiting Exoplanet Survey Satellite) transiting planet-hosting GK stars followed up with the HARPS-N (High Accuracy Radial velocity Planet Searcher for the Northern hemisphere) at TNG (Telescopio Nazionale Galileo) spectrograph within the Global Architecture of Planetary Systems (GAPS) programme. Methods. We derived stellar kinematic properties, atmospheric parameters, and abundances of 18 elements. Depending on stellar parameters and chemical elements, we used methods based on line equivalent widths and spectral synthesis. Lithium line measurements were used as approximate age estimations. We exploited chemical abundances and their ratios to derive information on planetary composition. Results. Elemental abundances and kinematic properties are consistent with the nearby Galactic thin disk. All targets show C/O<0.8 and 1.0<Mg/Si<1.5, compatible with an interpretation of silicate mantles made of a mixture of pyroxene and olivine assemblages for any rocky planets around them. The Fe/Mg ratios, with values of ~0.7–1.0, show a propensity for the planets to have big (iron) cores. All stars hosting very low-mass planets show Mg/Si values consistent with the Earth values, thus demonstrating their similar mantle composition. Hot Jupiter host stars show a lower content of O/Si, which could be related to the lower presence of water content. We confirm a trend found in the literature between stellar [O/Fe] and total planetary mass, implying an important role of the O in shaping the mass fraction of heavy elements in stars and their disks. Conclusions. Exploring the composition of planets through the use of elemental abundances of their hosting stars holds promise for future investigations, particularly with large samples. Meanwhile, the detailed host star abundances provided can be employed for further studies on the composition of the planets within the current sample, when their atmospheres will be exploited.