  In this paper we study isotopy classes of closed connected orientable
surfaces in the standard $3$-sphere. Such a surface splits the $3$-sphere into
two compact connected submanifolds, and by using their Heegaard splittings, we
obtain a $2$-component handlebody-link. In this paper, we first show that the
equivalence class of such a 2-component handlebody-link up to attaching trivial
$1$-handles can recover the original surface. Therefore, we can reduce the
study of surfaces in the $3$-sphere to that of $2$-component handlebody-links
up to stabilizations. Then, by using $G$-families of quandles, we construct
invariants of $2$-component handlebody-links up to attaching trivial
$1$-handles, which lead to invariants of surfaces in the $3$-sphere. In order
to see the effectiveness of our invariants, we will also show that our
invariants can distinguish certain explicit surfaces in the $3$-sphere.
