To help prevent the sagging of the bottom shelf, I would run two 2x4 along the front and back edge of the rack under the bottom shelf and then put casters in the 2x4. But this might take away some of the aesthetic aspects of it. In that case, I would definately put one in the middle so the bottom shelf isn't stressed too much.
It doesn't necessarily have to be thick, just strong. Some 1x4 would be fine; you'd probably only need maybe 4 or 5 evenly spaced along the back - every 4-6 inches would be overkill. But it would be better for it to be too strong than too weak.Now that Mat mentioned the sag the melamine would have over time I think a thick backing material nailed or screwed every 4 to 6 inches on each level should support the shelf from sagging.