Cosmic strings are topological defects possibly formed in the early Universe, which may be observable due to their gravitational effects on the cosmic microwave background radiation or gravitational wave experiments. To this effect, it is important to quantitatively ascertain the network properties, including their density, velocity, or the number of strings present, at the various epochs in the observable Universe. Attempts to estimate these numbers often rely on simplistic approximations for the string parameters, such as assuming that the network is scaling. However, in cosmological models containing realistic amounts of radiation, matter, and dark energy, a string network is never exactly scaling. Here, we use the velocity-dependent one-scale model for the evolution of a string network to better quantify how these networks evolve. In particular, we obtain new approximate analytic solutions for the behavior of the network during the radiation-to-matter and matter-to-acceleration transitions (assuming, in the latter case, the canonical Λ cold dark matter model) and numerically calculate the relevant quantities for a range of possible dark energy models.