We present the first analysis of the Euclid Early Release Observations (ERO) program that targets fields around two lensing clusters, Abell 2390 and Abell 2764. We use imaging data from the Visible instrument (VIS) and the Near-Infrared Spectrometer and Photometer (NISP) to produce photometric catalogs for a total of ∼500 000 objects. The imaging data reach a typical depth of 5 σ in the range 25.1–25.4 AB in the NISP bands and 27.1–27.3 AB in the VIS band. Using the Lyman-break method in combination with photometric redshifts, we searched for high-redshift galaxies. We identified 30 Lyman-break galaxy (LBG) candidates at z > 6 and 139 extremely red sources (ERSs), most of which likely lie at lower redshift. The VIS imaging is deeper than the NISP imaging, which means that we can routinely identify high-redshift Lyman-break galaxies at about a magnitude of 3, which reduces contamination by brown dwarf stars and low-redshift galaxies. The difficulty of spatially resolving most of these sources in 0′′ . 3 pix−1 imaging means that it is difficult to distinguish between galaxies and quasars. Spectroscopic follow-up campaigns of these bright sources will help us to constrain the bright end of the ultraviolet galaxy luminosity function and the quasar luminosity function at z > 6, and it will constrain the physical nature of these objects. Additionally, we performed a combined strong- and weak-lensing analysis of A2390, and we show that Euclid will contribute to constraining the virial mass of galaxy clusters better. We also identify optical and near-infrared counterparts of known z > 0.6 clusters in these data. These counterparts exhibit strong-lensing features. This establishes that Euclid can characterize high-redshift clusters. Finally, we provide a glimpse of the ability of Euclid to map the intracluster light out to larger radii than current facilities, which enables us to understand the cluster assembly history better and to map the dark matter distribution. This initial dataset illustrates the diverse spectrum of legacy science that is possible with the Euclid survey.