I believe you should start with the theory of tempered distributions, which are the linear functionals $\phi:\mathcal S(\mathbb R^n) \to \mathbb C$ where $\mathcal S(\mathbb R^n)$ is the Schwartz space on $\mathbb R^n$, i. e. the $C^\infty$ functions on $\mathbb R^n$ which are bounded together with all their derivatives.
You can get more intuition in $\mathcal S'$, since the tempered distributions behave pretty much as functions. In fact, every $f\in L^p$ is a distribution, via $$f(g) = \int fg$$ for every $g\in\mathcal S$. You can take a derivative $\partial$ of a distribution $\phi$ via $$\partial \phi(f) = -\phi(\partial f),$$ or the Fourier transform via $$\hat\phi(f) = \phi(\hat f\ ).$$ A good reference is Folland's Real Analysis book, Chapter 9.
