MINCE: I. Presentation of the project and of the first year sample

Context. In recent years, Galactic archaeology has become a particularly vibrant field of astronomy, with its main focus set on the oldest stars of our Galaxy. In most cases, these stars have been identified as the most metal-poor. However, the struggle to find these ancient fossils has produced an important bias in the observations - in particular, the intermediate metal-poor stars (-2.5 < [Fe/H] <-1.5) have been frequently overlooked. The missing information has consequences for the precise study of the chemical enrichment of our Galaxy, in particular for what concerns neutron capture elements and it will be only partially covered by future multi object spectroscopic surveys such as WEAVE and 4MOST. Aims. Measuring at Intermediate Metallicity Neutron Capture Elements (MINCE) is gathering the first high-quality spectra (high signal-to-noise ratio, S/N, and high resolution) for several hundreds of bright and metal-poor stars, mainly located in our Galactic halo. Methods. We compiled our selection mainly on the basis of Gaia data and determined the stellar atmospheres of our sample and the chemical abundances of each star. Results. In this paper, we present the first sample of 59 spectra of 46 stars. We measured the radial velocities and computed the Galactic orbits for all stars. We found that 8 stars belong to the thin disc, 15 to disrupted satellites, and the remaining cannot be associated to the mentioned structures, and we call them halo stars. For 33 of these stars, we provide abundances for the elements up to zinc. We also show the chemical evolution results for eleven chemical elements, based on recent models. Conclusions. Our observational strategy of using multiple telescopes and spectrographs to acquire high S/N and high-resolution spectra for intermediate-metallicity stars has proven to be very efficient, since the present sample was acquired over only about one year of observations. Finally, our target selection strategy, after an initial adjustment, proved satisfactory for our purposes.