Well, since no one has given a proper answer I will describe the regularity for the two equations I am familiar with. Maybe someone else knows how to answer in the case of the wave equations. For the sake of simplicity, lets assume Dirichlet boundary conditions in both cases, although this does not really affect the answer.
Poisson's Equation $\Delta u = f$:
This is an elliptic PDE. The usual way to attack the regularity of these is to use the Elliptic Regularity Theorem. A very basic form of this says, for example:
If $\Omega$ is a "nice" open set in $R^n$, $f \in H^k(\Omega)$ and $u$ satisfies $\Delta u = f$, then $u \in H^{k+2}(\Omega)$.
This implies a variety of generalizations to more general equations. For example, if $\Psi \in L^{\infty}(\Omega)$, $f \in L^2(\Omega)$ and $u \in L^2(\Omega)$ satisfies $\Delta u + \Psi u = f$, then we know that $\Delta u = f - \Psi u$. But the RHS is in $L^2(\Omega)$ by the assumptions on $u$, $f$ and $\Psi$. Thus $u \in H^2(\Omega)$. Also notice that the need to know a priori that $u \in L^2(\Omega)$ is not really a restriction since it is trivial by Lax-Milgram that $u \in H^1(\Omega)$. Moreover, if $f \in C^{\infty}(\Omega)$, then $f \in H^k(\Omega)$ for every $k > 0$. Thus if $\Delta u = f$ we know that $u \in H^{k+2}$ for every $k > 0$. Then the Sobolev Embedding Theorem implies that $u$ is also smooth. In general, you should expect solutions to be classical if $f \in H^k(\Omega)$ with $k + 2$ large enough so that the Sobolev Embedding implies that $u \in C^2_b(\Omega)$ These are the type of things to keep in mind when dealing with elliptic PDE.
Heat equation $u_t = \Delta u$, $u(0, x) = f(x)$:
This is a prime example of an evolution problem (it is also parabolic). These tend to have the property that the operator on the RHS ($\Delta$ in this case) generates a continuous (sometimes even analytic) semigroup (I recommend Roger & Renardy's book for an accessible introduction). It turns out that $\Delta$ generates an analytic semigroup $e^{t\Delta}$ on $L^2(\Omega)$ so the time regularity comes pretty much for free. Moreover, if $T(t)$ is an analytic semigroup with generator $A$ then $T(t)$ maps into the domain of $A^k$ for every $k > 0$. In the case of $A = \Delta$, this means that $e^{t\Delta}$ maps into $H^k(\Omega)$ for every even $k > 0$. Thus by applying the Sobolev Embedding Theorem, $e^{t\Delta}$ maps into $C^{\infty}(\Omega)$. Therefore, for every $f\in L^2(\Omega)$, the solution $u(t) = e^{t\Delta}f$ is a strong solution on $(0, \infty) \times \Omega$ and is immediately smoothed. So for this type of equation you get both existence and regularity if you can show that the operator on the RHS generates an analytic semigroup. A similar approach works for the Schrodinger equation, which is hyperbolic, so the main thing to notice here is the "evolution form" $u_t = Au$. So for these the solution will almost always be classical
I have never studied the wave equation, so in this case I have no idea.
Perhaps you should study some more advanced analysis, since that's when Frechet derivatives come up. A good (and legally free) reference is Applied Analysis by John Hunter and Bruno Nachtergaele.
After that, perhaps Analysis by Elliott H. Lieb and Michael Loss? It's more advanced, so be sure you understand Hunter and Nachtergaele first.
For more intense partial differential equations, UC Davis' upper division PDE's courses are available online too (with homework and solutions) when Nordgren taught it. There is Math 118A: Partial Differential Equations and 118B.
There are probably more advanced (free) references out there, but these are the ones I use...
Addendum: For references on specifically functional analysis, perhaps you should be comfortable with Eidelman et al's Functional Analysis. An introduction (Graduate Studies in Mathematics 66; American Mathematical Society, Providence, RI, 2004); I've heard good things about J.B. Conway's Functional Analysis, although I have yet to read it...
Best Answer
Regularity is one of the vague yet very useful terms to talk about a vast variety of results in a uniform way. Other examples of such words include "dynamics" in dynamical systems (I have never seen a real definition of this term but everyone uses it, and it vaguely means the way a system changes over time) or "canonical" (roughly meaning that with just the information given, a canonical choice is the most obvious choice) in more algebraic contexts.
In general, more regularity means more desirable properties.
Typically this means one or several of the following:
Furthermore, the issue of regularity is often isolated in proofs, so e.g. if you want to prove existence and uniqueness of a classical solution (i.e. "strongly" differentiable function that actually satisfies a PDE in exactly the way that it is posed) to a PDE, you first show existence and uniqueness of a weak solution which is often much easier (and allows to use tools from, for example, Hilbert space theory which are not available in most cases when looking for strong solutions). And only then you proceed to show that this solution is actually much more regular, for example, it actually possesses not only weak but even strong derivatives. A typical way to achieve this is using the Sobolev embedding theorems which tell you that if a function has sufficiently many weak derivatives, or rather, sufficiently regular weak derivatives (i.e. lies in a Sobolev space $W^{k,p}$ with large enough $k$ and large enough $p$), then it can be identified with a function that is actually strongly differentiable. This statement is called the "second part of the Sobolev embedding theorem" in the article linked above and is nicely summarised in the statement $$ W^{k,p}(\mathbb{R}^{n}) \subset C^{r, \alpha}(\mathbb{R}^{n}) $$ under suitable conditions on the "regularity coefficients" $k,p$ (and the dimension $n$).
Note also that these results are not limited to the full space $\mathbb{R}^{n}$, but also to domains $\Omega \subset \mathbb{R}^{n}$ that satisfy certain conditions (which, funnily, are also often referred to as "regularity" conditions, for example that the domains boundary $\partial \Omega$ can be represented by a "smooth enough" function, say a function in $C^{1}$, and then we write $\partial \Omega \in C^{1}$).
In this way, you can sometimes solve PDEs in the classical sense by making a detour into the realm of weak solutions.
Hope that helps!
EDIT: A nice example of this "weak + regular $\Rightarrow$ strong" at work can be found in the answer to this post: When can one expect a classical solution of a PDE?