The Blue Mountains is a solid ride, as Retrobyte says, but keep in mind you can easily take the bike on trains part way (apart from CityLink regional trains that require boxing). There will be no issues as long as you aren't going with the peak flow.
If you're in Randwick, Centennial Park will be your local. There's a nice 3.5 km circuit with various side roads with some rises that are good for intervals. Weekends are busy, but weekday mornings will be quiet. Some groups will ride there early morning or late afternoon.
After exploring Centennial Park, I would recommend something like dropping down to Clovelly along Clovelly Road and then following the coast through Bronte, Tamarama, Bondi, up to Watson's Bay, and Vaucluse, past Nielsen Park is a nice ride. Then turn left up New South Head Road and do a reverse of the outbound route. It's certainly a good route to reflect on global inequality while enjoying the view
You can also drop down New South Head Road and up again through Cranbrook and Victoria Roads but a bit of navigation through Bondi Junction is required, so best to get the lay of the land first.
The main thing will all of these routes is that there will be many sections with parked cars, so make the call to claim the lane, or drop to jogging pace and scan every parked car to make sure they aren't about to throw open the door.
And as a day ride, have a look at the World Champs course around Wollongong. You could do something like catch the train to Thirroul, ride the shared path along the coast, up Mt Keira, go up to the Mt Keira lookout, around to Mt Kembla, and down, plus a bit of the race course through the bottom of the escapement at Wollongong, and back to Thirroul. My only advice would be to avoid Princes Hwy/Crown Street as much as possible due to there not being any shoulder. A few minutes extra zigzagging on back streets is much more pleasant. There are enough options for looping to make it as hard as you want.
Enjoy!