Why Are Aussies So Hated?

How long have you got?
How far are you prepared to walk? The further you walk from the main tourist points the better the experience.
Katoomba is the place where all the tourist trips go. The train from Sydney takes 2hrs and costs about $20. There are dozens of tourist bus trip options from Sydney or else hire a car. It is about 100km / 2hrs +- from Sydney by any option.
This gives a flavour of what's there
https://www.explorerbus.com.au/places/171/bushwalk-giant-stairway-federal-pass.html it will be very busy with hundreds of tourists, especially on the 998 steps down to the valley.
I much prefer Wentworth Falls for a short visit. The walks start about 1 1/2 miles from the station with an occasional bus service. https://www.bestsydneywalks.com/wentworth-falls/ or via Charles Darwin track https://www.bestsydneywalks.com/charles-darwin-walk/
With a car the more remote areas are accessible. The Grosse Valley north of Blackheath or Mount Victoria is remote and stunning. Walks into the valley are 300 metres down and 300 metres up, best done with someone having bushwalking and first aid skill.
In July (mid-winter) we spent a night at the Hydro Majestic at Medlow Bath. An expensive hotel with an expensive restaurant but worth it for the sunset and sunrise views over the valley from our bedroom.
My all time favourite bit of the mountains is Newnes campsite in the Wollemi National Park (not the council campsite). It is 3 1/2 hours drive from Sydney. We stayed in December 2016 with no other people but lots of kangaroos, wallabies, wombats, goannas, lyre birds and noisy friar birds. The walk from the road to the Glow Worm Tunnel and back (4-5 hrs) was amazing. https://blog.nationalparks.nsw.gov.au/visit-the-glow-worm-tunnels-at-wollemi-national-park/