VELOcities of CEpheids (VELOCE) I. High-precision radial velocities of Cepheids

This first VELOCE data release comprises 18,225 high-precision RV measurements of 258 bona fide classical Cepheids on both hemispheres collected mainly between 2010 and 2022, alongside 1161 additional observations of 164 other stars. The median per-observation RV uncertainty is 0.037 km/s, and some reach 0.002 km/s. Non-variable standard stars characterize RV zero-point stability and provide a base for future cross-calibrations. We determined zero-point differences between VELOCE and 31 literature data sets using template fitting and measured linear period changes of 146 Cepheids. Seventy six spectroscopic binary Cepheids and 14 candidates are identified using VELOCE data alone and are investigated in detail in a companion paper (VELOCE II). Several new insights into Cepheid pulsations were obtained, including: a) the most detailed description of the Hertzsprung progression by RVs; b) the identification of double-peaked bumps in the RV curve; c) clear evidence that virtually all Cepheids feature spectroscopic variability signals that lead to modulated RV variability. We identified 36 such stars, of which 4 also exhibit orbital motion. Linear radius variations depend strongly on pulsation period and a steep increase in slope of the $\Delta$R/p versus logP-relation is found near 10d, challenging the existence of a tight relation between Baade-Wesselink projection factors and pulsation periods. We investigated the accuracy of RV time series measurements, v$_\gamma$, and RV amplitudes published in Gaia's DR3 and determined an average offset of 0.65 \pm 0.11 km/s relative to VELOCE. We recommend adopting a single set of template correlation parameters for distinct classes of large-amplitude variable stars to avoid systematic offsets in v$_\gamma$ among stars belonging to the same class. Peak-to-peak amplitudes of Gaia RVs exhibit significant (16%) dispersion compared to VELOCE. [abridged]