First of all, for me, it was the "Columbia Science Museum" with the "Art" part very much a secondary non-issue. Which is the opposite of the actual situation as in retrospect, the Science Museum was almost an afterthought in the combined enterprise.
There was a small lot behind Gibbes Planetarium where we would generally park, and take the brick walkway around the planetarium to the Science Museum front door. Right inside was the greeting desk which doubled as a gift kiosk. The only thing I can definitely recall being on sale there were Radiometers which were essentially light bulbs with solar-powered windmills inside. They also had various brochures and free premiums. The one I remember best, and which I am sure I still have an example of around somewhere was a small wallet card which had a table giving your weight on all the planets (and the sun).
If you walked past the greeting desk straight down the hall and steps all the way to the back door, you would come out in a small arboretum, no bigger than a couple of patios, planted with a variety of local plants, all described with placards. I think there was also a small pond, though I don't recall any fish.
If instead of going all the way to the back, you turned left, you would be in the main hall of the museum, which had a number of exibits, some semi-permanent, and some which changed from time to time. The semi-permanent exibits were a mad scientist's Jacob's Ladder, and a Foucault's Pendulum in a lucite cage which demonstrated something or other about the rotation of the earth. The jacob's ladder was my favorite, as it was "interactive" in the sense that it had a button you could push to turn it on. Watching the sparks climb the gap, and hearing the distinctive sizzle was something I found endlessly fascinating.
Off of the main hallway to the rear, was the museum's nature area where they had a bank of glass fronted cubicles with live examples of various South Carolina snakes, lizards and bugs. They also had a charmingly low-tech teaching device which had some sort of electrical wire, which if you toched it to the right answer to the wildlife question would illuminate a small bulb.
If at the front desk, you turned right, you would be in the small planetarium wing of the museum which housed the entrance to the planetarium, and a few other exhibits most of which changed from time to time. One which didn't change was the computer. I call it a computer, actually it was a piece of a computer, the "front panel" and some other parts if I recall correctly. Now the computer on which I'm typing this is much more powerful than the Science Museum computer, even if they had the whole thing there and running, but it's not nearly as impressive. The Science Museum computer was positively resplendent with cryptically labeled lights and toggle switches, and they let us flip any switch we wanted to! You know how in any old movie with a computer they always show the lights blinking on and off and a tape drive moving back and forth? They may not have had a tape drive (which were miserable to work with as I learned painfully later), but the lights and switches made up for it. In fact, I suspect on some level that my fascination with that partial computer combined with a number of other factors led me into programming..
If you walked down the main hall at the Science Museum past the nature room, you would come to the entrance to the Art Museum (or you could enter the main Art Museum door from the street). The Art Museum was basically the place your mother made you go after you had seen the Science Museum. It was set up in an old two story house with a fairly large one story addition on the back side. Since I didn't care much, my memory is pretty hazy, but I think they had a core collection with various exhibits on loan rotating in from time to time. I seem to remember that the bulk of the displays were in the rear, with the upstairs being reserved for especially uninteresting stuff like doll collections. Of course there were always a certain number of statues and paintings of naked ladies which were nice, but at which you could only glance briefly if with your mother. They always seemed to have some antique chairs, carefully roped off to keep them from being sat on, and enough nooks and crannies to do some running and hiding.
I can recall being interested in a particular show at the Art Museum only twice. Once they had part of the King Tut treasures as a visiting exhibit, something that drew huge crowds, and another time when I was in high school, they had a hologram exhibit which became a class field trip for most of the city schools (and was a rare good use of the upstairs space). Apparently it was a bit premature to call holograms an art form however, as they have kind of fallen by the wayside as a true "artistic medium".
Apparently the Art Museum had been feeling cramped for quite a while, and with the closing of most of the Main Street retail district, a lot of prime real estate became available downtown. In 1998, the Art side of the museum moved to Main Street, and the Science Museum and planetarium were simply closed with the buildings eventually being used by the USC Campus Police. There's probably an old warehouse somewhere in Columbia with a box labeled "computer parts" holding the marvelous space-aged Science Museum computer front panel..
UPDATE 21 June 2011: Added two pictures [at top] of the Science Museum from an old Chamber of Commerce promotional book. First kids queueing by the planetarium and then kids learning about nature in the back garden.