We loved it here and wish that more visitors to Australia would take the time to visit Adelaide and SA! It's baffling why it doesn't have greater popularity. Adelaide is a compact and very civilised city - a great contrast to our stay in Sydney which was vibrant, but intense and mega-busy. Adelaide is more laid-back and that's no bad thing for visitors. The central area is laid out on a strict grid system bounded by 4 "Terraces" and bus/tram services are free within this area. There is also free wifi within the CBD, which is really useful.
The city is a great mix of modern buildings and surviving Victorian architecture and the Museum of South Australia is excellent, ditto the eclectic SA Art Gallery next door to it on North Terrace. The cricket ground is lovely and my husband has decided that if he ever watches a Test Match again in Oz (he must be a masochist...) he would prefer to watch it in Adelaide. Eating out seems to be an Adelaide preoccupation: there must be more cafés/restaurants here than anywhere else: you will never go hungry and it will be a rare cuisine that isn't represented. We ate in a different place every night (will do some reviews) and thanks to the responders to our recent TA question about places to eat. So many restaurants and only one stomach: we abandoned all thoughts of having lunch.
The timing our visit has coincided with the record breaking heat wave and that has curtailed some of our activities: 45 degrees is just too damned hot to be doing tourist stuff outside, apart from early morning. Even the evenings in the city have been baking. Our hotel (Crowne Plaza - very good) has been handing out bottles of water to guests as they leave the hotel, imploring them to keep hydrated. We took the tram to Glenelg and hit the beach for a few hours one morning, where it was several degrees cooler than the city. The Botanical Garden is great (their Museum is very interesting) and the Central Markets were well worth a visit. The central shopping area along Rundle Mall is very good, with loads of restaurants nearby along Rundle St and Gouger St. We have developed a liking for Coopers Pale Ale!
We would have liked to do a day trip on the Murray River but unfortunately the timings for us didn't fit: something for next time. Overall, we unreservedly recommend Adelaide as a great place to visit, although avoiding a period of 45 degree heat will be preferable! Next stop, Perth...