Adelaide is a beautiful, easy to navigate city. If you are staying in the city centre, you can easily walk to the zoo (and see the pandas sit around doing not much), botanic gardens, Adelaide oval and along the river Torren's. A walk along North Terrace will take you past parliment house, government house, the war memorial, the museum, art gallery, Adelaide Uni and the botanic gardens. Adelaide city centre is surrounded by a ring of parklands, so you are never far from somewhere tranquil. The Japanese gardens off South Terrace are beautiful.
If you are in Adelaide on a Sunday, take a trip to Port Adelaide. For about $3 you can do a boat ride along the not very scenic Port river. You are however pretty much guaranteed to see the Port river dolphins. Every time I have been along the Port river I have seen the dolphins as they come up to nearly every boat that uses the river, big or small. The info centre on Commercial street can provide you with maps for self guided walking tours through the Port, going past some of the most beautiful buildings you will see in Australia. The Fishermans wharf markets are open on Sunday also if you want to buy something you never knew you needed. Port Adelaide is also home to the Maritime museum, Railway museum (heaven for little kids) and the Aviation museum.
From Adelaide city centre you can catch a tram to Glenelg for a day at the beach. Adelaide's beaches are clean and safe for swimming. The media will scare you with stories of sharks but you wont see one. I never have and I have been a regular to Adelaides beaches for a very long time.
If you want to get out of the city for a while, hire a car and drive to one of the nearby wine regions. The Barossa Valley, McLaren Vale and the Adelaide Hills are all within an hours drive of Adelaide city centre. South Australia has excellent quality food and wine that you can afford.
For wildlife you go to the Adelaide Hills. Cleland wildlife park allows a close up view of many Aussie animals and you can even hold a koala. Warrawong santuary at Heathfield is free to visit, full of native animals. For a fee, you can even stay overnight.
Have a great holiday. almost 7 years ago