Can someone recommend a textbook or review article that covers exactly solved models in statistical mechanics, such as the six- or eight-vertex models? If there is literature at the undergraduate level, that would be ideal. I'm only familiar with Baxter's classic text on the subject, but this is tough reading for an undergraduate student.
How familiar are you with exact solutions to the 2D Ising model? If you don't know that forwards and backwards, I would probably start with some textbooks that cover that material in depth. This would teach you some of the basics of transfer-matrix techniques and some other tricks about estimating eigenvalues, the thermodynamic limit, and so on. The book that I like is Plischke and Bergersen's Equilibrium Statistical Physics (3rd Edition). Plischke and Bergersen devote a whole chapter to solving the 2D Ising model, and work through it in a lot of detail. One prerequisite is some Quantum Mechanics - the solution that they present uses Pauli matrices and some creation and annihilation operators. They also do an Ising model on a fractal lattice (and on a chain, of course), both examples that can be solved exactly. This textbook won't teach you six- or eight-vector models directly, but if you work through it, Baxter will be a lot easier.
There are scanned copies floating around on the net - just google it if you want a preview.
I would like to add that it is very important to know what you can find out without knowing the exact solution. Kardar's book "Statistical physics of fields" teaches that in a very engaging way. To get quickly started, I suggest the topics of Scaling theory, and real space renormalization in 1-d ising model. Your motivation in keeping up with Baxter will grow due to this book.