I'm supposing you mean for $g, h > 0$. Associated with a surface bundle, there is a homomorphism of $\pi_1(\Sigma_h)$ to the outer automorphism group of $\pi_1(\Simga_g)$. \pi_1(\Sigma_g)$. This is equivalent (with slight low-genus modifications) to a homotopy class of maps into the modular orbifold, Teichmüller space modulo the mapping class groups. The group of homeomorphisms of a surface homotopic to the identity is contractible, so these bundles are determined up isomorphism that acts as the identity on the base and on one fiber.
The conjugacy problem for the mapping class group is solved, using either the theory of pseudo-Anosov homeomophisms or automatic group theory, and either of those tools allows you to solve isomorphism up to bundle maps that are the identity on the base. Peter Brinkmann's program xtrain, which you can find online, computes the dilatation constant, which is typically enough to distinguish conjugacy classes in the mapping class group. Snappea, also available online, will usually distinguish homeomorphism classes of the 3-manifolds obtained by an element of the mapping class group (with exceptions that can be analyzed). This will also distinguish conjugacy classes, by looking for homoemorphism preserving a cohomology class.
The action of the mapping class group of the base on bundle maps seems trickier, and I don't think I know an immediate answer of classifying them. The troublesome cases would be where the image of the surface group in the mapping class group is not a quasi-isometric map of groups.
A classification of homeomorphism types would include the special case when the surface bundle is induced from a map of the base to a circle, so the bundle comes from a 3-manifold that fibers over a circle. 3-manifolds can fiber in many different ways, so not all homeomorphisms in these cases are fiber preserving, and the homeomorphism classification for these particular cases is solvable, but it gets into a complicated theory that won't usually work for 4-manifolds. I'm not sure what's known about surface fiber bundles over surfaces that fiber in multiple ways, apart from these.
One other point: the fundamental group of such a 4-manifold has an action on $S^1$, namely, the circle at infinity to the fibers. The action is faithful if the monodromy of the bundle is faithful. In these cases, the isomorphism class of the 4-manifold I believe is determined by the subgroup of homeomorphisms of the circle, up to conjugacy.
For $h > 1$, there is always some branched cover of the base surface so that when you pull the bundle back to the branched cover, there is a section of the bundle, the map to the outer automorphism group of the fiber lifts to the automorphism group, and the fundamental group of the 4-manifold is a semi-direct product.
I'm not an expert in these, and I'm sure there is more that is known.