Context. Classical Cepheids are the most important primary indicators for the extragalactic distance scale. Establishing the precise zero points of their period-luminosity and period-Wesenheit (PL/PW) relations has profound consequences on the estimate of H₀. Type II Cepheids are also important distance indicators and tracers of old stellar populations. Aims. The recent Data Release 2 (DR2) of the Gaia spacecraft includes photometry and parallaxes for thousands of classical and Type II Cepheids. We seek to review the classification of Gaia DR2 Cepheids and to derive precise PL/PW for the Magellanic Clouds (MCs) and Galactic Cepheids. Methods. We adopted information from the literature and the Gaia astrometry and photometry to assign DR2 Galactic Cepheids to the classical, anomalous, and Type II Cepheids classes. Results. We reclassified the DR2 Galactic Cepheids and derived new precise PL/PW relations in the Gaia passbands for the MCs and Milky Way Cepheids. We investigated for the first time the dependence on metallicity of the PW relation for classical Cepheids in the Gaia bands, finding inconclusive results. Conclusions. According to our analysis, the zero point of the Gaia DR2 parallaxes as estimated from classical and Type II Cepheids seems likely to be underestimated by ∼0.07 mas, which agrees with recent literature. The next Gaia data releases are expected to fix this zero point offset to allow eventually a determination of H₀ to less than 1%.