I lived in Sydney for almost a year a while ago(1987) so I couldn't recommend an invidual restaurant. But 2 generalities. I had the best Indian food I've ever had, probably because of the British influence. Terrific Chinese because of all the asians. Fabulous seafood with different fish you don't see here. Make sure to have Syndey rock oysters and balmain bugs. The bugs are a crustacean-almost like a crab. They look prehistoric. Go to the Sydney waterfront where all the seafood comes in. Next best thing to Tsukijii market in Tokyo.
North of Sydney was a park, maybe called Whairakei(sp?) where there was a large enclosure with kangaroos and emus. You buy bags of food there and they all come and eat out of your hand. Great experience. Especially seeing all the joeys in their pouches. Don't know if it's still there.
Hope you're renting a car in Tasmania. Beautiful country to drive around. Or did you mean Indiana
Shrimp on the barbie isn't what we think of as grilled fish. What they call grilled means putting a griddle or flat steel plate on top of a grill, usually gas, and cooking it. More like a hot saute. Our aussie friends thought it strange when I cooked shrimp or lamb chops directly on the grates over the fire. Liked it though.
David Jones, the Harrod's of Sydney, had a terrific food hall in their basement. Again, I don't know what it's like now or if David Jones is still there, but it would be a must see. Try a lamington if you see it(coconut topped piece of cake) and pavlova. Fish and chips.
When in Brisbane, try to get out to the Barrier Reef. Don't know if you scuba dive but if so it will be the experience of your life. At least try snorkeling. If I think of anything else, I'll let you know.