The highly debated effect of metallicity on the absolute magnitudes of classical Cepheid variables needs to be properly quantified for determining accurate and precise distances based on their Leavitt Law. Our goal is to obtain homogeneous optical and near-infrared light curves of Milky Way Cepheid variables complementing their already collected high-resolution spectroscopic metallicities as part of the C-MetaLL survey. Together with Gaia parallaxes, we investigate period-luminosity-metallicity relations for Cepheid variables at multiple wavelengths. We present homogeneous multiband (grizJHKs) time-series observations of 78 Cepheids including 49 fundamental mode variables and 29 first-overtone mode variables. These observations were collected simultaneously using the ROS2 and REMIR instruments at the Rapid Eye Mount telescope. The Cepheid sample covers a large range of distances (0.5-19.7kpc) with varying precision of parallaxes, and thus astrometry-based luminosity fits were used to derive PL and PW relations in optical Sloan (griz) and near-infrared (JHKs) filters. These empirically calibrated relations exhibit large scatter primarily due to larger uncertainties in parallaxes of distant Cepheids, but their slopes agree well with those previously determined in the literature. Using homogeneous high-resolution spectroscopic metallicities of 61 Cepheids covering -1.1<[Fe/H]<0.6dex, we quantified the metallicity dependence of PL and PW relations which varies between -0.30+/-0.11 (in Ks) and -0.55+/-0.12 (in z) mag/dex in grizJHKs bands. However, the metallicity dependence in the residuals of the PL and PW relations is predominantly seen for metal-poor stars ([Fe/H]<-0.3dex), which also have larger parallax uncertainties. The modest sample size precludes us from separating the contribution to the residuals due to parallax uncertainties, metallicity effects, and reddening errors. While this Cepheid sample is not optimal for calibrating the Leavitt law, upcoming photometric and spectroscopic datasets of the C-MetaLL survey will allow the accurate derivation of PL and PW relations in the Sloan and near-infrared bandpasses, which will be useful for the distance measurements in the era of the Vera C. Rubin Observatory's Legacy Survey of Space and Time and upcoming extremely large telescopes.