  We introduce our Radial Velocity Survey for Planets around Young stars
(RVSPY), characterise our target stars, and search for substellar companions at
orbital separations smaller than a few au from the host star. We use the FEROS
spectrograph 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. We determine for all
target stars their basic stellar parameters and present the results of the
high-cadence RV survey and activity characterization. We achieve a median
single-measurement RV precision of 6 m/s and derive the short-term intrinsic RV
scatter of our targets (median 22 m/s), which is mostly caused by stellar
activity and decays with age from >100 m/s at <20 Myr to <20 m/s at >500 Myr.
We discover 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 detect 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 find 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 analyse the TESS photometric
time series data and find significant periodicities for most of the stars. For
11 stars, the photometric periods are also clearly detected in the RV data. We
also derive stellar rotation periods ranging from 1 to 10 days for 91 stars,
mostly from TESS data. From the intrinsic activity-related short-term RV
jitter, we derive the expected mass-detection thresholds for longer-period
companions.
