From the airport to the city centre, you can take the bus or get a taxi if you haven't arranged transfers as part of your holiday, and once in the city, the public transport is excellent value for money - particularly if you buy a day pass and use it a lot.
It's a great idea to arrange car hire in Adelaide, and you can get cheap rates if you do it before you travel, and there is plenty to see on a tour of the area, including many of Australia's finest vineyards - if you do any tasting avoid driving, because the Australian police are renowned for not giving much leeway to drunk drivers.