Given that $a=3$, $b=2$, and $c=\sqrt{19}$,

By the cosine rule, we have $\cos C = \frac{a^2 + b^2 - c^2}{2ab}$

$= \frac{9 + 4 - 19}{2 \times 3 \times 2} = -\frac{1}{2}$,

Since $0 < C < \pi$, we have $C = \frac{2\pi}{3}$, which implies $\sin C = \frac{\sqrt{3}}{2}$.

Therefore, the area $S$ of $\triangle ABC$ is:

$S_{\triangle ABC} = \frac{1}{2}ab\sin C = \frac{1}{2} \times 3 \times 2 \times \frac{\sqrt{3}}{2} = \boxed{\frac{3\sqrt{3}}{2}}$.