Does a "functional" always takes in a function and spit out a number?
This is what a professor said in class a long time ago but now I am studying Frechet derivative and a claim was made that a linear functional is $Ax: V \to W$. Can $W$ be anything other than a field i.e. a functional space?
A functional is a map from a vector space $V$ to its underlying scalar field $F$. So not quite. For example the norm on $\Bbb{R}^3$ is a functional because it maps the vector $(x,y,z)$ to the real number $\sqrt{x^2+y^2+z^2}$.
If you have a function space (a vector space of functions), then a functional is a map from the vector space (functions) to the scalar field (numbers). For example on $L^1(\Bbb{R})$ the integral which maps $f$ to $\int_{\Bbb{R}} fdx$ is a functional.
A functional can map functions to numbers, and this is the context you'll see it mostly. Or other vectors to other scalars. Functionals defined on exotic vector spaces are still functionals.