There are a few, not many, books on hyperbolic equations. You might have a look to that of S. Benzoni-Gavage and myself: Multi-dimensional hyperbolic partial differential equations. First order systems and applications, Oxford Mathematical Monographs, Oxford University Press (2007).
A basic fact of hyperbolic systems of PDEs is that the Cauchy problem is well-posed in both directions of time. Therefore the regularity of the solution cannot be improved as time increase, contrary to the parabolic case. Also, this implies that such systems cannot be recast as gradient flows; instead, some of them can be reformulated as Hamiltonian system (say, if the semi-group is reversible).
That said, there exists nevertheless some regularity properties. On the one hand, the singularities are polarized. This means that the solution is smooth along non-characteristic directions, and most of (but not all) the solution is smooth even in characteristic directions. Let me take the example of the wave equation $$\partial_t^2u=\Delta_x u$$ in ${\mathbb R}^{1+d}$. Then the wave-front set is invariant under the bi-characteristic flow $$\frac{dx}{dt}=p,\qquad\frac{dp}{dt}=-x.$$ A by-product (which can be proved directly by an integral formula of the solution) is that if the initial data $u(t=0,\cdot)$, $\partial_tu(t=0,\cdot)$ is smooth away from $x=0$, then the solution is smooth away from $|x|=|t|$. However the wave-frontset approach tells you much more.
On another hand, the decay of the initial data at infinity implies some space-time integrability of the solution. These properties are not directly related to hyperbolicity. They are consequences of the dispersion. In the case of the wave equation, this is the fact that the characteristic cone $|x|=|t|$ has not flat part. Such integrability statements are know as Strichartz-like inequalities.
Finally, the ODE point of view is adopted by Klainerman, Machedon, Christodoulou and others, mixed with Strichartz inequalities, to prove the well-posedness of the Cauchy problem for semi-linear hyperbolic systems, like Einstein equations of general relativity.