What should a PDE/analysis enthusiast know?

I think Besov spaces and their connection to wavelets/multiresolution analysis is an especially useful tool.

If you want to talk about PDE's in a modern context, you need to think carefully about the function spaces the solutions live in. A lot of people study function spaces just enough to get by - looking at Sobolev spaces, maybe Hölder spaces and stop there - but Besov spaces are more general and very useful for studying PDE's in nonsmooth domains or with rough coefficients. You can pose PDE's on domains that are a fractal, or things like that. I think they give a better intuitive picture of smoothness of functions, embeddings, trace theorems, etc than Sobolev spaces alone.

A great reference is, Hans Triebel, "Theory of Function Spaces II"

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...

One nice idea along these lines is the Leray-Schauder Fixed Point approach to non-linear elliptic existence problems.

Roughly speaking, for example, if I would like to solve the quasilinear problem

$a_{ij}(x,u,Du)D_{ij}u + b(x,u,Du) = 0$

$u = \varphi$ on $\partial\Omega$,

say, I can set up a map which takes a function $v$ and sends it to the unique solution $u$ of the linear problem

$a_{ij}(x,v,Dv)D_{ij}u + b(x,v,Dv) = 0$

$u = \varphi$ on $\partial\Omega$.

Call this map $T$. Obviously I need to know existence for the linear problem. Then, by considering this as a map (not a linear map) between appropriate Banach spaces (for which I need some regularity for solutions of the linear problem) I can see that a solution to the quasilinear problem is precisely a fixed point of $T$.

Googling Leray-Schauder fixed point theorem seems to turn up a nice set of notes by Leon Simon on the subject. Also Chapter 11 of Gilbarg and Trudinger explains the method well.

The method is so-named because of the Leray-Schauder fixed point theorem - an abstract fixed point theorem (that is to say it is just about maps between Banach spaces and not specifically about PDE) which, under certain conditions gives a fixed point of $T$. The task of verifying the conditions under which the theorem is applicable is one of gaining apriori estimates for solutions of the quasilinear problem, so this approach is a good way to see the need for the apriori philosophy in (elliptic, at least) existence problems.