  Let $\xi$ be a stable Chern character on $\mathbb{P}^1 \times \mathbb{P}^1$,
and let $M(\xi)$ be the moduli space of Gieseker semistable sheaves on
$\mathbb{P}^1 \times \mathbb{P}^1$ with Chern character $\xi$. In this paper,
we provide an approach to computing the effective cone of $M(\xi)$ after
showing that it is a Mori dream space for all $\xi$. We find Brill-Noether
divisors spanning extremal rays of the effective cone using resolutions of the
general elements of $M(\xi)$ which are found using the machinery of exceptional
bundles. We use this approach to provide many examples of extremal rays in
these effective cones. In particular, we completely compute the effective cone
of the first fifteen Hilbert schemes of points on $\mathbb{P}^1 \times
\mathbb{P}^1$.
