In contrast to the other answers, I would like to mention that it is possible to compute rigorously the value of the critical temperature of the two-dimensional Ising (and Potts) model, without computing explicitly the free energy (which is in any case not possible for general Potts models). In the Ising case, this has been known for a long time, and there are various proofs. The first result valid for all Potts models is this one. Note that it also establishes sharpness of the phase transition (that is, the fact that correlation decay exponentially fast as soon as $\beta<\beta_c(q)$).
In the case of the two-dimensional Ising model, one can also compute the critical temperature for general doubly periodic lattices. A proof can be found here.