The South Australia Visitor & Travel Centre can supply information on B&Bs and home stays around the state. The rates given here are rack rates, or what the hotels hope they'll get on a good day -- you can often get a room for much less, especially by booking directly through the hotel's website.
Plan Ahead -- If you plan to be in town during the Adelaide Festival, make sure you book accommodations well in advance. The town can be packed during Christmas and New Year's, so it's wise to book well in advance then, too.
In North Adelaide -- This leafy and attractive suburb across the river is an interesting place with nice architecture and a great restaurant strip. It's about a 10-minute bus ride from the city center.