To clarify, I realize that we will have to FLY from Adelaide to Sydney.

You really wouldn't want to have a car in Sydney, public transport is the way to go. You can't park anywhere and the kind of places tourists generally want to go: Taronga Park Zoo, Bondi, the Aquarium, Art Gallery of NSW are all quite spread out, but well serviced by buses, ferries and trains.

Adelaide is a good city to walk around, it's fairly compact, but if you wanted to go up into the Hills a car would be useful.

Canberra is not the easiest city to navigate on public transport because it's so spread out. There are, however, very regular buses that go from the city centre to the cultural precinct on the other side of the lake. So a car isn't essential.

I agree with what Susan7 has said, but I think you'll miss out on some fabulous views and the great nature reserves around Canberra if you don't have a car. While it may be easy enough to get from Civic (the city center) to the museums on the other side of Lake Burley Griffin, there's not a good way to get out to Namadgi National Park or Tidbinbilla Nature Reserve without a car. You'd probably also want a car to get up Mt. Ainsley or Red Hill for the views of the city (although my husband regularly used to walk up Red Hill for exercise!) There are a number of roundabouts to deal with in Canberra, but it's the first place I ever drove on the left and I found it easy due to the lack of heavy traffic.

I have toured Canberra without a car and it's is not a great option. Using buses will limit what you can get done in a day and as longhorn says you will miss some of the beautiful nature reserves. There is an explorer bus that might be a good option. It's about A$30 for the day ticket or 50 for 2 days
http://www.canberradaytours.com.au/AboutExplorerBus.php

There is also some good links for exploring Canberra attractions here
http://www.squidoo.com/canberra-australia

If you have any Canberra questions feel free to contact me. I have taken groups to Canberra twice a year for about 15 years. Not much I have not seen down there and I always enjoy it.

I agree about the car being almost essential in Canberra. If you just wanted to do the Galleries/High Court/Houses of Parliament and were happy to walk, you could take a taxi/ bus there, but you'd be limited to the bus schedules for everything else. And there's a lot of other interesting things to see, which don't necessarily take a lot of time.

Lovely city - was there a week or so ago & really enjoyed our 3 days.

I also felt Canberra was an easy city to drive in.

It has been years since I have been in Adelaide, but at the time there was a free bus that looped around the city center, taking one around the shopping area, to the South Australian Museum, and near the excellent Central Market.
Adelaide FREE 99C Loop Service:
http://www.adelaidemetro.com.au/routes/cityservices.html
There are also numerous trams taking you to other parts of the city, such as Glenelg, Adelaide's beach side resort suburb.
Kangaroo Island is an easy trip from Adelaide. Again, with this you wouldn't need a car. We booked an overnight tour through Sealink Ferries. We caught their bus from central Adelaide (not far from our hotel) to the ferry, just in time to spend the night on Kangaroo Island, then spent the entire next day touring the island. It was wonderful. We saw beautiful scenery and loads of wildlife. Returned to Adelaide the following evening.
http://www.sealink.com.au/tours
We did rent a car for a couple of days to tour some of the wine regions (mainly McLaren Vale, Adelaide Hills, and Barossa Valley). In the wine country, we overnighted at Warrawong Wildlife Sanctuary, which was fun.
I wouldn't be so nervous about driving on the other side of the road; there is not much traffic once you leave the cities. Australian drivers in general seem polite, largely law-abiding, and in no great hurry.
As others have stated, it is easy to get around Sydney without a car. You have bus, ferry, train, light rail, and even monorail to choose from. If you are reasonably fit and stay in a central location, it is easy to walk to many of the city sights, such as the opera house, the botanical garden, the Rocks, the Harbor Bridge, the Art Gallery of New South Wales. Ferries drop you at Toronga Zoo, or Manly Beach, or the Sydney Aquarium (you can take other forms of transit, as well, but the ferry rides are more scenic and fast).
We took a tour around Canberra. If you prefer not to drive, consider taking day tours.

Diamantina - very much appreciate your tips. Thank you for taking the time to do that!

Hi jranger,

How set are you on visiting those three areas? Another option could be to hire a car in Adelaide for the last few days, then drive the famous Great Ocean Rd to Melbourne, then fly from here to Sydney.

I have never been to Canberra so I may be bias in cutting it out.

As the previous posts say, Sydney there is definitely no need for a car. I have found everything is in walking distance or very easily accessible.

I live in Adelaide, and don't have my licence - there is no problem getting around. While your husband is doing business you could shop down Rundle Mall or Rundle St in the CBD, visit the Central Markets and Chinatown, go down to the river for a relaxing afternoon or visit the Museum or Art Gallery, which are next to each other in the CBD. You could catch a tram which goes to Glenelg (popular seaside area) from the main street in the CBD which come every I think 7 minutes during peak times and 15 minutes other times. To visit the Adelaide Hills, you could catch public transport (there is good public transport to some areas) or you could go on a tour bus. The only restrictions would be the wine regions McLaren Vale and Barossa Valley, these would need to be reached by car or there are numerous tour options (which if you are doing a lot of tastings would be worthwhile!). In these areas you can get world class meals if you are a foody. My favourite of the two is McLaren Vale, and it is closer to the CBD.