How long is your trip? Although it looks as if you want to concentrate on the North East corner, so better to just explore that thoroughly.
In general try and spend as few days in those cities as possible, as there's a good deal of gorgeous landscape to take in inbetween.
If you're into outdoor activities (hiking, canoeing, whitewater rafting, diving, snorkelling, riding etc) I'd also consider how much your budget can stretch to include them as Queensland's a great place to just enjoy running around in.
In and around Sydney:
- walking around the heads
- relaxing in the Chinese/Japanese gardens
- Take a tour of the opera house, it's fascinating!
- Check out Ku ring gai national park
On the way to Brisbane:
- Byron bay of course
- Surfers paradise if you're up for that kind of thing
Brisbane to Cairns:
- Fraser Island
- Airlie beach and the whitsundays
Cairns:
- Diving or snorkelling to see the only living thing visible from space - the Great Barrier Reef
- Atherton tablelands (catch a train or cable car there for something a bit different)
- Mission beach for some cassowary spotting
North of Cairns:
- Cape Tribulation
- Daintree rainforest
- Mossman Gorge
...If you can get up to Cape York that'd be great!
I will point out that I've only been to Brisbane and Rockhampton in Queensland. I don't know the area beyond what I hear from other travellers and my own research cos I was hoping to get to Cairns this year.
A few last points - Queensland has a quarter of Australia's indigenous population. Take that as an opportunity to learn more about our oldest civilisation - there are lots of organisations and tour companies that can help you, but try and pick ones that are Aboriginal owned or run to benefit communities.
Lonely Planet's 'Aboriginal Australia' is an incredible resource. Unfortunately they don't seem to have been able to publish a new edition since 2001 so the contact details are occasionally out of date, but it's still a good guide on where to look for such experiences and most of the time you can google the current details.
Also make sure you consider the environment - we don't say 'slip slop slap' (Slip on a shirt, slop on sunscreen and slap on a hat) for nothing! Make sure you always have plenty of water on you, pay attention to the flags and lifesaver's advice on beaches, mind fire regulations at camp sites and 'take nothing but photos, leave nothing but footprints' at our National Parks.
Patronise accommodation and tour operators that are ecofriendly. Tourism Tropical North Queensland have a ‘planet safe partnership’ that recognises such businesses. Also look for the Eco Certified logo – businesses display the ‘tick logo’ have been certified by Eco Tourism.