We study the intrinsic large-scale distribution and evolution of seven ionized metals in the IllustrisTNG magnetohydrodynamical cosmological simulation. We focus on the fractions of Cii, Civ, Mgii, Nv, Neviii, Ovi, and Siiv in different cosmic web structures (filaments, haloes, and voids) and gas phases (warm-hot intergalactic medium, hot, diffuse, and condensed gas) from z = 6 to z = 0. Our analysis provides a new perspective to the study of the distribution and evolution of baryons across cosmic time while offering new hints in the context of the well-known missing baryons problem. The cosmic web components are here identified using the local comoving dark matter density, which provides a simple but effective way of mapping baryons on large scales. Our results show that Cii and Mgii are mostly located in condensed gas inside haloes in high-density and low-temperature star-forming regions (rho(gas)/(rho) over bar (bar) greater than or similar to 10(3), and T less than or similar to 10(5) K). Civ and Siiv present similar evolution of their mass fractions in haloes and filaments across cosmic time. In particular, their mass budgets in haloes in condensed phase (rho(gas)/(rho) over bar (bar) greater than or similar to 10(3), and T less than or similar to 10(5) K) are driven by gas cooling and star formation with a peak at z similar to 2. Finally, our results confirm that Ovi, Neviii, and Nv are good tracers of warm/hot and low-density gas at low redshift (rho(gas)/(rho) over bar (bar) greater than or similar to 10(3), and T greater than or similar to 10(5) K), regions that are likely to contain most of the missing baryons in the local Universe.