The preceding "toy models" focus mainly on algebraic properties of quantum state-spaces. Students who prefer a more geometric and/or PDE perspective on quantum dynamics may find it instructive to consider classical dynamical systems, which (historically) have always been a rich source of quantum dynamical "toy models".
<b>Classical:</b> The state-spaces of classical dynamical systems typically are endowed with both a symplectic structure and a metric structure. Canonical examples ("toy problems") include the Bloch equations and the Landau-Lifshitz-Gilbert equations: both have $S^2$ as their state-space; the symplectic and metric structures are natural; and in both cases dynamical trajectories unravel according to canonical fluctuation-dissipation relations.
<b>Quantum:</b> $C^n$ is the state-space; the symplectic and metric structures are natural (that is, Kählerian); and dynamical trajectories unravel according to canonical fluctuation-dissipation (that is, Lindbladian) relations.
In this geometric dynamical framework, what are often regarded as "spooky mysteries" of quantum mechanics are seen to be similarly present in classical theories; the main change is that global spectral theorems that apply in $C^n$ appear classically as local theorems involving dynamical flow.
These ideas are scattered throughout in the literature of math and physics, but there is at present no single textbook that develops this unified classical/quantum dynamical point of view. This for me defines " <i>[a book that you would like to write][1]</i>" (in Gil Kalai's phrase); further references to the literature can be found in [that post][1].
In the meantime, a good "toy" exercise is to write down the symplectic and metric structures, and the Hamiltonian and Lindbladian potentials, that describe the Bloch equations in geometrically natural terms. In particular, an illuminating exercise is to derive via classical geometric methods the results that Charles Slichter's <i>[Principles of magnetic resonance][2]</i> derives quantum mechanically in Section 3.2, equation 3.7.
If completing this exercise is difficult, then a good question to ask oneself is "What aspects of classical dynamics am I not appreciating properly?" This question leads naturally to the study of metric and symplectic structure, and in particular, to the classic writings of Mac Lane, Arnold, and Abraham and Marsden. In turn, these writings do much to illuminate how quantum dynamics works, from a modern geometric point-of-view.
The preceding is *ONE* class of "Toy Models of Quantum Mechanics", at any rate!
[1]: http://mathoverflow.net/questions/54232/a-book-you-would-like-to-write/54364#54364
[2]: http://books.google.com/books?id=zgnrRkaIhFoC&pg=PA66