The Blue Mountains are certainly worth it- one of my favourite spots in all of Sydney, and I’m a Sydneysider. Shop around for accommodation though… it can get pricey. The Katoomba YHA is actually really good quality accommodation, especially for those on a budget. If you have a car, look further afield than Katoomba (which is tourist central) and look for accommodation around places like Blackheath or Lawson. It’s only a short drive on the main road to all the places to sightsee. Plus, there are other more beautiful (in my opinion!) places to see in the Mountains, other than Echoes Point- for example, Wentworth Falls, Govett’s Leap and Leura is stunning.
I would urge you to look beyond the city and visit the outback as well. Suggestions like Kings Canyon, Uluru and other gorges along the way are great. There are plenty of tour groups that drive you around there, plus there are options to fly directly to Uluru, stay and fly out again. Road trips are great fun (I did one), however they take ages, so don’t do a road trip if you’re on a limited time in Australia and you want to pack as much as you can into your trip. On the plus side, road trips allow you to see more, such as Stanley’s Chasm, Simpson’s Gorge, Glen Helen Gorge, Katherine Gorge. These places are overwhelmingly beautiful.
Even though I don’t have firsthand experience of the top Northern Territory, I would strongly recommend Kakadu and the Kimberleys, firstly because they are under threat of mining and the option to visit this pristine wilderness may not be available in coming decades. As a side note, I tend to associate the Northern Territory with self-guided road trips, so as to direct yourself and not get cheated on by inflated tour guides.
Queensland is stunning. however the humidity and unreliable weather in the summer months (Nov-Feb) can make it a bit hit and miss. Flooding and bushfires in the same state, with jellyfish and bluebottles in the water… my best advice is to see the top north in the winter months. However, it is important to see Cairns, Green Island and the Great Barrier Reef before coral bleaching becomes a thing. Kuranda and other such nature reserves, and the Daintree, are beautiful, but July.
While Sydney is iconic with the Opera House and the Harbour Bridge, you can do all that in a day. You can walk from Circular Quay and through the historic Rocks part of Sydney (maybe go on a weekend and see the Rocks markets which are great) then wander on up to the Opera House and then take a stroll through the Botanic Gardens, which have picturesque views of the harbour. You could do all of this in a few hours to be honest. I take the walk on my lunch break.
I really think you should invest time in Melbourne. It’s a beautiful city with free transport in the CBD and excellent food/shopping precincts. From Melbourne, you can get a taste of Ballarat/Bendigo, which are historic and would give you a taste of rural and regional Australia, if you don’t get a chance to head out to Uluru. There’s also access to the Twelve Apostles and the Penguin Parade on Phillip Island, which are also an absolute treat.