Adding to the list others have contributed...
If you have kids or if you've never seen Oz animals up close then Cleland Wildlife Park is well worthwhile.
Adelaide has some nice beaches, which you will appreciate since you will be hear at the height of summer.
The Governor Hindmarsh Hotel is a good live music venue. Womadelaide is March 5-8 in 2010, so you'll miss that.
Sailing in Adelaide can be pretty cheap, and some operators will teach you as you go.
Not sure if your partner cycles. If so, the ride from the city to McLaren Vale via the rail trail is recommended for a nice day out. The linear park (ie, the banks of the Torrens River) is a nice ride too.
The Museum has a fine collection of aboriginal day-to-day artifacts (most collections concentrate on the ceremonial artifacts, which is pretty but not as illuminating). Also a fine collection from PNG, including weapons and head-hunters's collected heads. The research portion of the museum is the best in Australia, and that shows in the collection. The museum holds some of the oldest fossils found, but be warned that early fauna is disappointingly small and basic.
The railway museum is Port Adelaide is fun if you have kids as they can actually go in the cabin. And it's huge. The aviation museum next door has some nice stuff, but currently lacks enough interpretive material so people outside of aviation can appreciate what they are seeing. The Maritime Museum is good if you have kids -- there's a real sailing boat inside.
The Port Princess tours the river each Sunday from the Port Markets (a junk/craft market). Easily the best way to see the Port River. The best market for wandering about in is the Gilles St Market (3rd Sundays).
Central Markets is a food market, and pretty much the only remaining true-to-life retail food market in Australia. Well worthwhile, but avoid peak shopping times. There's a Chinatown annex, but don't bother as it's nothing like Sydney. You can also shop for veg at the North Arm Markets on Sunday, which are cheap.
The SA Art Gallery has its own sweet charm, but isn't in the same class as Melbourne's.
The Zoo is excellent. It's in two parts: a compact city zoo and a open range zoo about an hour out of town. There's a range of special tours across summer, and you might see more on those than during the heat of the day.http://adelaide-in-photos.blogspot.com/
is a local cyclist who goes to a lot of events, you might want to see what you can pick up from there. As
Handorf is a complete tourist trap, and yet so very nice. Crafts and arts for the girls, Germanic food for the lads (or vice versa if you wish). The Adelaide Hills have a lot of very nice small towns.
Someone runs a tour of the second-hand fashion shops. I'll ask my better half if you PM me for details.
South Australia specialises in niche foods. Everyone knows about the wineries, and a trip to a historic one like Seppeltsfield is well worthwhile. But there's also Maggie Beer's farm, Haighs Chocolates, a few cheesemakers, and so on.
If you want to get out of town then SA offers geography. Usually it's a long way away -- the salt lakes, Coober Pedy, Flinders Ranges. Closer to hand is Kangaroo Is. January is a peak period, but KI is well worthwhile visiting for 2-3 days. You can do a day ferry+bus trip, but it is very hurried.
Finally, I should warn you that Adelaide is a town of many small pleasures. This makes it an absolute delight to live in. But perhaps less accessible to people trying to jam in a lot of high-energy attractions. For example, my home is in Semaphore. As a local I might have breakfast at Froot Cafe, walk the Semaphore Jetty, have a go at a sideshow game on the foreshore, buy a coffee from Noonies Kiosk, wander down Semaphore Road, have dinner at Sarah's Sister's restaurant. Although I do hear there will be an international class bike race on the week you are here