For me, I built a brass orrery, but that probably doesn't help others trying to display it.
I like the idea of a wide thin plaque/base, with brass rods coming out supporting the planets all in a row. Labels with planets names and details could then be added to the base of each rod. Crafts stores and hobby stores often have a "K+S Metals" stand where brass rod can be purchased, as well as a variety of pre-carved wood plaques. As well as that you'd want a drill, some glue, something to cut the rod (either pliers or hacksaw, or a tube-cutter if using tube instead of rod) and if the base is unfinished wood then probably also a spray-can of wood-stain and a spray-can of varnish (also sold at craft stores and hobby shops).
For a more modern version of the same presentation idea, a crystal clear acrylic base and silver rods? Glass-like clear plastic rods? Glossy black plastic base?
For a top-of-the-line presentation, perhaps use ponoko.com for laser-cutting - upload a design to be laser cut out of any color of acrylic, with labels laser-etched into the surface, and they'll mail you the finished parts (though this would probably cost about $40). They also laser cut and etch some woods. Hmm, I should do this and make the resulting laser-cut bases available for purchase via Ponoko. Not sure i'll have free time for that any time soon though.
The holes in the base of the planets only go in about 1/8th inch, so to mount firmly on rods it might be easier to drill the hole a bit deeper.