If you want to avoid getting killed by the local fauna, enjoy Australia's two isolated patches of civilisation, Melbourne and Sydney. Stay away from open water and nature. You could visit Uluru or a meteor crater while you're there, but you'll need to watch out for deadly drop-bears and giant spiders that erupt out of the ground and swallow people whole while snarling like dogs.
New Zealand on the other hand, is like a graceful and comfy armchair the size of Japan. You'll be here in autumn/winter, it's pretty temperate, so if you're going hiking you'll need warm gear and flexible plans.
In the warmer, creatively named North Island, there's a good hike called the Tongariro Crossing. At altitude and very exposed though. There's a whole lot of geothermal sightseeing in the central North Island, along with touristy Maori cultural experiences.
You'll probably land in Auckland in the North of the Island. Auckland is full of and surrounded by docile volcanoes. There are the Waitomo caves and Hobbiton from Lord of the Rings south of Auckland, moving south you get to the geothermal area, lake Taupo (a supervolcano), Mt Tongariro, Mt Ruapehu (Skiing, check when the season usually starts). The southern part of the North Island is pretty docile and farmy. The capital Wellington is pretty cool, good to visit as lots of good stuff is in walking distance. Surprisingly good cultural scene, craft beer everywhere, you can get a ferry to the South Island.
The imaginatively named South Island is the most beautiful place on Earth. You may have extremes of amazingness in the US, but the South Island is everything you could need packaged conveniently. There's a good walk though the Abel Tasman National Park at the north end, but it might not be the best time of year for it. There are whale watching trips out of Kaikoura on the way south to Christchurch, which is recovering from a savage earthquake in 2011. There are some cool things around the town which has generally had good disaster recovery results in the city.
If you go all the way south to Dunedin, it's also a good cultural city, huge indie music cred, but small by world standards. There are some cool bits of nature around there, north of Dunedin there's an area of surreal round boulders on the beach at Moeraki, the coastal area south of Dunedin is called the Catlins and is pretty cute.
The west coast of the South Island is narrow and steep, generally pretty remote, and has the people New Zealand is most proud of. Several mining areas, classic pubs. Don't know what you'd do there though, I've never been. In between are the Southern Alps, and the roads through them are pretty awesome. There are some scenic train rides available in both Islands too, worth considering as a lot of the nature is not packed with actual destinations, it just sits there being epic.
Tips - We don't tip, just pay the bill.
People are generally very friendly by world standards, don't freak out if you have instant friends unless you're obviously getting played as a tourist. Oz and NZ are remote places, we love visitors. If you get talking, you'll find we love individual Americans, have a pretty cynical view of "America", and can't tell your political parties apart.
Our gangs are generally not psychos, they're old school compared to US ones. You might not see them at all anyway.
Be careful of your stuff, but you're not likely to get robbed. We only started locking our doors in the last 20 years.
There are few places to avoid. NZ society is not as polarised as most of the world. There are some suburban areas you wouldn't want to look out of place in, but they're small, spaced out all over the country and you won't see them unless you stop off in remote small towns or get on the wrong city bus. Same goes for Oz AFAIK.
Being a New Zealander, I haven't done much of the above or been to all those places. We go overseas as soon as we can afford it as a rite of passage. I'm happy to tell you anything else I know, feel free to give a rundown of your plans and interests, all the above is pretty mainstream.
FWIW I live near Wellington and don't go out enough to know what's cool right now, 20 years of mixing bands and breaking up fights used up all my love of the nightlife.
Good of you to make the effort to come down here though, respect.
Fear? I know not fear. There are only moments of confusion. Some of them are deeply stamped on my memory and a few will haunt me forever. Hunter S. Thompson
Eschew the monumental. Shun the Epic. All the guys who can paint great big pictures can paint great small ones. Ernest Hemingway