Despite Adelaide's reputation as just a big country town, you won't find anywhere easily accessible with dark skies without driving out to the countryside - difficult without a car.
In the city, you could try for some night shots along North Terrace (around the State Library, SA Museum, SA Art Gallery, and Uni Adelaide/UniSA precinct. Another option could be around the National Wine Centre and Bicentenial Conservatory (next to the Wine Centre in the Adelaide Botanic Gardens. Looking west from Hackney Road (near the corner of Botanic Rd) could be an interesting city location for a sunset/moonset shot, With the National Wine Centre building to the left, the National Wine Centre vineyard in front, and the Conservatory behind. (click
here)
For an indoor city location (if you have time during the day), the Adelaide Central Market is a fabulous place to visit. Great food and produce, but a great photo location as well.
http://www.adelaidecentralmarket.com.au/
Adelaide beaches also make for great sunset locations. Glenelg, Brighton, and Henley beaches all have jetties. Glenelg is a simple tram ride from the city centre.
The beaches and coast in the southern suburbs are lovely but a little harder to get to without a car. Hallett Cove Conservation Park is on the coast (just next to the Hallett Cove Beach train station). A work friend of mine has taken some great shots there (for example
this)
Richard (Adelaidian now living in country NSW)