Exactly solvable models in Statistical mechanics by Baxter is the place to look.
The other answers have already pointed the infinite range Ising model and 1D Ising model as exactly solvable (although the latter has phase transition at zero temperature). Besides the infinite and the 1D case, the Ising (and more generally Potts) model is also solvable on a Bethe lattice, as discussed extensively in the book cited above.
There are also offshoots of these models with additional features: e.g., Sherrington-Kirkpatrick model is the infinite range Ising model with random couplings, which is useful for understanding spin glasses.
However, the specific features of phase transitions in these models differ, and thus the model of choice depends on what one is looking for.
Remark: As more exotic case of exactly solvable models one could mention soluton of Kondo and Anderson impurity models by Bethe ansatz.