Context. The occurrence rate and period distribution of (giant) planets around young stars is still not as well constrained as for older main-sequence stars. This is mostly due to the intrinsic activity-related complications and the avoidance of young stars in many large planet search programmes. Yet, dynamical restructuring processes in planetary systems may last significantly longer than the actual planet formation phase and may well extend long into the debris disc phase, such that the planet populations around young stars may differ from those observed around main-sequence stars. Aims. We introduce our Radial Velocity Survey for Planets around Young stars (RVSPY), which is closely related to the NaCo-ISPY direct imaging survey, characterise our target stars, and search for substellar companions at orbital separations smaller than a few au from the host star. Methods. We used the FEROS spectrograph, mounted to the MPG/ESO 2.2 m telescope in Chile, to obtain high signal-to-noise spectra and time series of precise radial velocities (RVs) of 111 stars, most of which are surrounded by debris discs. Our target stars have spectral types between early F and late K, a median age of 400 Myr, and a median distance of 45 pc. During the initial reconnaissance phase of our survey, we determined stellar parameters and used high-cadence observations to characterise the intrinsic stellar activity, searched for hot companions with orbital periods of up to 10 days, and derived the detection thresholds for longer-period companions. In our analysis we, have included archival spectroscopic data, spectral energy distribution, and data for photometric time series from the TESS mission.Results. For all target stars we determined their basic stellar parameters and present the results of the high-cadence RV survey and activity characterisation. We have achieved a median single-measurement RV precision of 6 m/s and derived the short-term intrinsic RV scatter of our targets (median 23 m/s), which is mostly caused by stellar activity and decays with an age from >100 m/s at <20 Myr to <20 m/s at >500 Myr. We analysed time series periodograms of the high-cadence RV data and the shape of the individual cross-correlation functions. We discovered six previously unknown close companions with orbital periods between 10 and 100 days, three of which are low-mass stars, and three are in the brown dwarf mass regime. We detected no hot companion with an orbital period <10 days down to a median mass limit of ∼1 M Jup for stars younger than 500 Myr, which is still compatible with the established occurrence rate of such companions around main-sequence stars. We found significant RV periodicities between 1.3 and 4.5 days for 14 stars, which are, however, all caused by rotational modulation due to starspots. We also analysed the data for TESS photometric time series and found significant periodicities for most of the stars. For 11 stars, the photometric periods are also clearly detected in the RV data. We also derived stellar rotation periods ranging from...