Of all the places you mention, LHI is truly a gem, often referred to as the Jewel of the Pacific and when I went there were flights from both Sydney and Brisbane so if you check whether that is still the case you could make a triangular trip, flying from Sydney to LHI and then from there after a few days fly to Brisbane, their supply ship which goes to and from from Yamba having a couple of cabins and so that could be another experience, getting you back to Northern NSW, not all that far from Byron Bay and then you could make your way south towards Sydney again visiting what places you want to, Julie raising a point about why Newcastle, Gosford, Illawarra and I will add even Canberra in summer for it can be quite hot with countryside looking rather brown.
For sure, there are some wonderful beaches around Newcastle just as there are in many locations along the NSW coast, north and south of Sydney, interspersed with plenty of national parks.
To get the best experiences, you may want to hire a car for a week between Byron Bay and Sydney or longer to first of all bypass Sydney to do some of the south coast, maybe even time to get as far south as Eden, doing some camping a great way to experience many beach and national park locations before returning to Sydney.
You need to also bear in mind that late December through January will be very busy as it is Australia's major holiday period so perhaps best t avoid that time and come in February though that is usually our hotest month and for camping, anytime from mid March through to mid May would be great.
Check out NSW National Parks to get detailed info on many locations.
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