We’re planning the big trip down under and can’t decide whether to do Australia and New Zealand together or just one. If we had about three weeks, is it mad to try both, or does it turn into a blur of flights? For those who’ve done both, how did you split it?
Three weeks for both sounds like a lot, especially with the flights between Australia and New Zealand. Maybe if you only want to see two or three places in each, it could work, but otherwise why not just choose one country and enjoy it properly? And if you really want to do two countries, maybe choose ones that are closer to each other so you don’t spend so much of the trip travelling.
given the size of australia, the flights required between destinations, and the sheer level of amazing things to see and do in both countries, I would definitely split them up! give each their own trip, so you can truly soak it all in!
Are you including your initial travelling time and the return home travelling time in the three weeks? That cuts out a few days on it’s own. (Don’t forget, when planning, to check if you lose or gain any time.You will if you are coming from the USA. And it effects follow on plans and booking dates) The best plan to make use of time can only be made if there is an idea of where you are coming from but If from the US fly into Sydney. Stay a couple of days, visit the Blue Mountains via a tour. Then fly to Uluru for three nights. Then fly to Cairns and spend a couple of days visiting the Great Barrier Reef. From there fly to Christchurch New Zealand South Island. Spend a week either driving the tourist loop or take a tour. Consider the Trans Alpine train between Queenstown and Christchurch at the beginning or end. Then fly to Auckland spend a couple of days and home.
If coming from Asia fly to New Zealand first. Then to Sydney , Uluru and Cairns. you might be able to fly home from Cairns.
If from Europe fly from Singapore to Christchurch NZ do the loop then fly to Cairns do the reef (fly to Sydney if you must 
then Cairns)then fly to Uluru for the rock ,fly or catch a tour back to Darwin if you want more natural outback experiences, and fly back to Singapore.
These options let you see the main attractions that people want to see and can be viewed and accessed fairly easily with limited time available. Make sure you check the time of year you re visiting. The Summer and Autumn months are good for New Zealand. Summer is not good for Outback Australia as it is too hot and rainy , as will Cairns. Autumn or Spring will be best times.
It is a lot of flying but the destination airports are small and quick to navigate. It is a lot to try to fit in visiting two countries in a few weeks and only have a glimpse . You could decide whether you want to see the natural beauty of New Zealand with fantastic scenery and see both islands in three weeks, or see one or two cities and have some outback glimpses in Australia. I would not recommend just visiting major cities in Australia unless you have a specific purpose or attraction to see. If you are coming from Europe or Asia you could do all of New Zealand and just have a stopover to see Sydney on the way back. You will have to come back again!