  The primary goal of this paper is to derive precise Fourier parameters of the
radial velocity (RV) curves for fundamental and first-overtone Galactic
Cepheids. For each star, we carefully selected RV measurements available in the
literature which yield the highest precision of Fourier parameters. We
performed a Fourier decomposition of the RV curves. We subtracted RV
modulations caused by binary motion and have removed other residual trends.
Finally, we have displayed and analyzed qualitatively the progressions found
for Fourier parameters. We applied a standard identification of their pulsation
mode based on their Fourier phase $\phi_{21}$. Our final sample includes 178
fundamental-mode and 33 first overtone pulsators, as well as 7 additional
Cepheids whose pulsation mode is uncertain or undetermined according to our
criteria. For the fundamental-mode Cepheids, we improved the precision of
Fourier parameters in comparison of previous results from the literature. We
are able to firmly identify V495 Cyg as a new first-overtone Cepheid. We
confirm first-overtone nature of several other stars. We also show that
$\alpha$ UMi should be definitely classified as a first-overtone pulsator. In 3
objects (VY Per, AQ Pup and QZ Nor) we found significant $\gamma$-velocity
variations that we attribute to spectroscopic binarity. Finally, the analysis
of the F-mode Fourier parameters up to 7th order reveals tight progression of
Fourier phases. For $P<10\,$day we find a well defined upper limit for the
Fourier amplitude $A_1$. The pulsation period coverage and the precision
obtained, in particular for Fourier phase $\phi_{21}$, will be useful for
studying the dynamics of Cepheid pulsations with the help of hydrodynamical
models. Further radial velocity measurements from modern high-resolution
spectroscopic instruments will be important to improve these results.
