When to use Fourier, Laplace and Z transforms?
It's natural consequence of applying a transform to a convolution relation. The output $y(t)$ of an (continuous-time) LTI system is described by a convolution integral :

$$y(t) = h(t)\star x(t) = \int_{-\infty}^{\infty} x(\tau) h(t-\tau) d\tau $$

And when you apply a Fourier transform on this relation, it turns out to be a multiplication in the transform domain such as:

$$ Y(j\omega) = \mathscr{F}\{ y(t) \} = \mathscr{F}\{ h(t) \star x(t) \} $$
$$ Y(j\omega) = \mathscr{F}\{ h(t) \} \cdot \mathscr{F}\{ x(t) \}  = H(j\omega)\cdot X(j\omega) $$

Similary you can also apply a Laplace transform on it :
$$ Y(s) = \mathscr{L}\{ y(t) \} = \mathscr{L}\{ h(t) \star x(t) \} $$
$$ Y(s) = \mathscr{L}\{ h(t) \} \cdot \mathscr{L}\{ x(t) \}  = H(s)\cdot X(s) $$

And exactly the same happens for discrete-time LTI systems.