If you're just getting around town, staying on North Terrace, and not planning on actually touring around, then I would agree that the car is probably more trouble and expense than it's worth.

I've never caught the bus to/from the airport, but the distance is so short that a taxi is pretty cheap, about $25 I think.

Adelaide in general is very relaxed and easy to get around. Buses seem to be pretty omnipresent, and there's not that much traffic in town so I'd imagine they tend to run on time; there's also a single tram line, and a train network which probably doesn't go anywhere you want to go.

North Terrace is a good place to stay; it's only busy by Adelaide standards, not by big city standards. And most of the interesting sights in town are lined up along it.