Visiting Aus this year!! Tips and Recommendations?😊

Hi! Me and my family are planning to visit this end of May (probably on the 20th) till mid June, we'll be staying for a minimum of 2 weeks. I have one 4 year old sibling and the other is 13, Im well aware that it is quite cold during that time of year, we did that on purpose :)

We'll mostly be staying in Sydney and Melbourne, but we'd also like to check out other places like the Ocean road or the Blue mountains and such since we wouldnt really want to stay inside the city throughout the duration of our stay, but things to do in those cities would be great as well! Im not sure if we could go anywhere on road further than 7 hours or so, but anywhere 'nature-y' near those areas would be nice to know.

Any ideas on how we can make the most of our stay? (considering we have a budget too) Tips and recommendations would help alot<3

Edit: what clothes should we wear? is a light jacket enough?