In this work, we present the results of an observational study of 2I/Borisov carried out with the 10.4-m Gran Telescopio Canarias (GTC) and the 3.6-m Telescopio Nazionale Galileo (TNG), both telescopes located at the Roque de Los Muchachos Observatory, in the island of La Palma (Spain). The study includes images in the visible and near-infrared, as well as visible spectra in the 3600-9200 Å wavelength range. N-body simulations were also performed to explore its orbital evolution and Galactic kinematic context. The comet's dust continuum and near-infrared colours are compatible with those observed for Solar system comets. From its visible spectrum on the nights of 2019 September 24 and 26, we measured CN gas production rates Q(CN) = (2.3 ± 0.4) × 1024 mol s-1 and Q(CN) = (9.5 ± 0.2) × 1024 mol s-1, respectively, in agreement with measurements reported by other authors on similar nights. We also obtained an upper limit for the C2 production rate of Q(C2) < (4.5 ± 0.1) × 1024 mol s-1. Dust modelling results indicate a moderate dust production rate of ∼50 kg s-1 at heliocentric distance rh = 2.6 au, with a differential power-law dust size distribution of index ∼-3.4, within the range reported for many comet comae. Our simulations show that the Galactic velocity of 2I/Borisov matches well that of known stars in the solar neighbourhood and also those of more distant regions of the Galactic disc.