If you’re there in February it will still be light in the evening. So make the most of that. So after work in the evening
1 go for a wander round the botanical gardens
2walk round circular quay and the rocks
3 take a ferry to manly, stay for dinner and ferry back when dark to see the city lights
4 take the ferry to watsons bay and walk the cliffs to Bondi
5 go to Bondi and do the walk to Coogee beach
6 take a ferry up the river to parramatta
7 walk across the harbour bridge then train back
get an opal card which covers ferries trains buses and light rail
For your 6 days definitely at least one day in the blue mountains, more if you like walking. The Opal card extends that far although the train would probably be quite slow.
I love Newcastle which is north of Sydney. This is also a day trip.
Spend some time at a beach - pick your favourite out of Bondi,Manly etc. Bondi has a very strong undertow so swim between the flags.
See if there’s anything on at the opera house that you fancy seeing.
There are a few different wildlife parks if you want to see native animals up close.
Do the walk the bridge - you might need to book this in advance. Be warned they breath test you so don’t drink much the night before.
Go up the Sydney tower for the view. It has a revolving restaurant too
Have a wonderful time