Virgo is the nearest galaxy cluster; it is thus ideal for studies of galaxy evolution in dense environments in the local universe. It is embedded in a complex filamentary network of galaxies and groups, which represents the skeleton of the large-scale Laniakea supercluster. Here we assemble a comprehensive catalog of galaxies extending up to ~12 virial radii in projection from Virgo to revisit the cosmic-web structure around it. This work is the foundation of a series of papers that will investigate the multiwavelength properties of galaxies in the cosmic web around Virgo. We match spectroscopically confirmed sources from several databases and surveys including HyperLeda, NASA Sloan Atlas, NASA/IPAC Extragalactic Database, and ALFALFA. The sample consists of ~7000 galaxies. By exploiting a tomographic approach, we identify 13 filaments, spanning several megaparsecs in length. Long >17h^-1^Mpc filaments, tend to be thin (<1h^-1^Mpc in radius) and with a low-density contrast (<5), while shorter filaments show a larger scatter in their structural properties. Overall, we find that filaments are a transitioning environment between the field and cluster in terms of local densities, galaxy morphologies, and fraction of barred galaxies. Denser filaments have a higher fraction of early-type galaxies, suggesting that the morphology-density relation is already in place in the filaments, before galaxies fall into the cluster itself.