I visited Adelaide back in 2006. Here is my top list:
--Walk around the Central Market. It's the largest open air food market in the southern hemisphere
--EAT. Adelaide is a fantastic food town.
--Barossa Valley Wine Tasting
--Clare Valley Wine Tasting.
--River Torrens Park Trail: Great, beautiful walking along the river in the circle of park around central Adelaide
--Glenieg Tram. Ride the tram from central Adelaide to the coast suburb of Glenieg.
--North Adelaide: Walk around this historic neighborhood for shopping and dining