Brit, now living in OZ.
I'd fly in and out of SYD, when you see the bridge/opera house you realise you're 'actually in Australia'
Sydney
Harbour (bridge climb)
Aquarium
Zoo (Amazing, even just for the views - consider roar and snore)
Icebergs
Bondi/coogee - for beaches
Be aware you/especially the kids will feel crap for the first couple of days, so include downtime.
Cairns
You can get a nice air bnb sleeps 7 with a pool for around $1.8k for 4 nights. (Over Easter)
Definitely book ahead peak season for Australian's and the rest of the world, so we attract lots of tourists.
Any GBR tour will be excellent - you will be sick on the way out if your even slightly prone to seasickness/motion sickness. So take that into account.
Lots of good day trips to the rain forest.
Brisbane/Gold Coast/Sunshine Coast
Noosa is beautiful and you can get amazing houses there, you could think about getting one on sunrise beach - absolutely breathtaking.
Also can commute from there to Australia Zoo, which is good but not a must do if you've done the one in Sydney. However you can only cuddle koala's in QLD not NSW.
It will be warm enough to swim in Easter, so plenty of beach days.
Teens will love surf lessons and paddle boarding through the noosa river surrounded by multi-million dollar homes.
Don't discount the GC, yes it's tacky but in a fabulous way. Stay in Broadbeach, not surfers. Kids will love the theme parks.
Sky tower is great at sunset.
Shopping is great and it will be going into our winter, so you'll be able to buy here to wear at home.
Take a day trip down to rick shores for a boozy adult lunch. Leave the kids on the beach !
Don't discount Brisbane, it isn't the boring city everyone thinks!!
Take into account OZ can be expensive. It will still be hot at Easter, especially up north. Theme parks/experiences are expensive and will quickly add up over the course of 3 weeks. So think about maybe 'gifting' some special experiences as Christmas gifts.
'Swim with sharks in the Sydney aquarium'
'Scale the harbour bridge'
'Cuddle a koala at Australia zoo'
Easter weekend, plan a 'staycation on those days' everything closes on Good Friday. You can eat out but at a 15/20% premium because it's a public holiday. Pubs/bars are closed. But you can eat at a restaurant, with a 2 hour limit.
Service/shops will be limited on Sunday/Monday too and often incur the surcharge. So staying in a nice house over that weekend would be a good idea.
One side note to think about is luggage. If you travel with a big carrier and get 30kg allowance. Be aware that the smaller internal carriers (tiger/ Jetstar - budget airlines or qantas/virgin) will only give you 23kg.... be mindful !!
Have an amazing trip OP, let me know if you have any Q's.