Uluru isn't very interesting in itself, IMO, but if you hire a car and go out to the Olgas (aka Kata Tjuta) as well, they're a lot more interesting (IMO); or you can take a bus tour to King's Canyon which is like a miniature Grand Canyon. Need to be good at walking for that though.
Blue Mountains are great for a day trip, walking around Echo Point (see the 3 sisters) and go to Scenic World - the funicular railway is best for going down, the cableway gets you back up again and the Skyway, with it's disappearing floor, is pretty spectacular! You can do as much or as little walking around that area; the board walkways have various routes to cut your walk down if you need to. We've had some good times there - saw a lyre bird in full flow once! Amazing creature.
If you like caves, then Caves aren't too far away either.
Tasmania is a great place - very like England! - and Hobart is lovely. Salamanca market is (or was when I was there some years ago) very interesting, full of all sorts of eclectic stalls! Richmond in Tasmania has the best preserved convict prison - it's not the biggest, but it's the best (and it's bloody eerie, especially the solitary confinement!) and they also have the oldest stone bridge in Australia in Richmond. There are some amazing waterfalls - Russell Falls in Mt Field National Park is spectacular if you like waterfalls.
Cairns is worth a visit if your children want to go out to the Barrier Reef - I get that you don't but what about the others? - and there are plenty of tours out of Cairns that go up to the Atherton Tablelands, further north to Cape Tribulation (where Captain Cook scraped the reef and his ship was grounded) which has one of the best beaches I've ever been to, it's a world heritage site, but when I went (ok in the late 1990s) there was horseriding available on the beach - no cantering mind!
I have no personal knowledge of Perth or WA but there is a beach somewhere around there called Monkey Mia where the dolphins come in very close.
Sydney is a very large city (of course!) but has its attractions if you want to see the Opera House, the harbour, walk the bridge, go to the Aquarium (or take a ferry across to Manly and go to the Manly Aquarium if you like smaller ones), also there is a wildlife park next to the Aquarium and you can get a deal on the tickets. Taronga Zoo is another but very big zoo - that's an all-day outing! Very hilly, and pretty expensive.
Canberra is dullsville from a scenic point of view but does have some great stuff to do! Although maybe your 2 might be a bit old for the dinosaur museum etc.
North of Brisbane is Australia Zoo - one of the best I've been to, especially for seeing wombats! Definitely the best wombat enclosure, and as we got there at opening time once, one of them was being taken for a walk on a harness and I got to pat her! 
. The Eumundi markets are ENORMOUS and probably another day's worth of visit all by themselves; and there are some great chocolate-box style villages up there - Montville is my favourite :)
If I think of anything else of interest I'll post again. :)