You should be able to bring the cats into the cabin of the plane with you. They must stay in a crate that has to fit under the seat in front. There are two of you so you will be able to bring a cat each onto the plane.
Quantas does not allow pets, but nearly all of the others do. This is the link to the United pet page. United flies into Melbourne, but I do not know where you are flying from though.
One tip, go to a health food store and get "rescue remedy", it's a herbal mixture that has an amazing calming effect on animals. It should take the edge off the stress that the cats will be going through. It's completely natural with no side effects at all.