We consider the two-dimensional two-component plasma, or Coulomb gas, consisting of $N$ positive and $N$ negative charges with logarithmic interaction. We introduce a suitable regularization of the interaction by smearing the charges over a small length scale $λ$, which allows us to give meaning to the system in the continuum at any temperature. We provide an expansion of the free energy in terms of the inverse temperature $β$, as $N \to \infty$ and $λ\to 0$. Doing so allows us to show that, for $β\geq 2$, the charges (for the most part) pair into neutral dipoles of very small size as $λ\to 0$. This complements the prior work of Leblé, Zeitouni, and the second author, which proved that this does not happen for $β< 2$, thereby implying a transition at $β= 2$ from free charges to dipole pairs. Moreover, we obtain an estimate on the size of linear statistics. The description in terms of dipoles is made via a decomposition into nearest-neighbor graphs of the point configurations, à la Gunson-Panta. This is combined with new energy estimates obtained via an electric reformulation of the interaction energy and a ball-growth method, which are expressed in terms of the nearest-neighbor graph distances only. In this way, the model is compared to a reduced nearest-neighbor interaction model, by showing the relative smallness of the dipole-dipole interactions.