Worked in the NZ for 5 years. Overall, I’d say it’s a little easier as an American to set up shop on NZ vs. Australia. Though (unless they changed some laws since I was there) an Australian residency permit allows you to work in NZ, but you need NZ citizenship to work in Australia.
Check out inz.govt.nz for your visa options. I haven’t researched in a few years (so things might have changed) but if you’ve just graduated, you likely qualify for a working holiday visa. A Silver Fern visa may also be an option. If you can find a suitable job without already having a visa, you can easily get a work visa (the job is the hard part because you need to convince an employer they should hire you, then your employer needs to convince immigration that there were no qualified Kiwis).
And if you’re planning staying longer than two years, there are residency visas/permits you can apply for. There’s a point system they use to be eligible (you’ll be hearing the Trump administration talk about how much better the NZ system is than our own). With a degree and your age, you should easily have enough points to qualify, though the process is slow and you’ve got to front a fair bit of money (In the $2-3k range is my memory).