It's been a few years since I lived in Auckland (and I was not active with records then) but depending on your transport options I'd highly suggest hitting the west coast - Piha, Karekare beaches. It's not gonna be swimming weather but they're pretty incredible places. E.g. Karekare is the remote-looking spot holly hunter's piano gets dropped off at the beginning of The Piano).
Auckland is generally good for accessibility to nature - coromandel's an example, but even closer, the Waitakere ranges and Hunua ranges have some incredible walks. Unfortunately people keep tracking in a disease that kills the few dozen ancient kauri trees left in the country so a load of walks are closed, but there are still some that won't threaten any (basically) gods. https://ourauckland.aucklandcouncil.govt.nz/articles/news/2018/5/kauri-dieback-closures-what-you-need-to-know/
In Queenstown I can recommend the Cow pizza place. I know you're probably booked up but I'd also say, Arrowtown is a drive outside of town but it's a cool bit of history that a visitor probably wouldn't be aware of - it's an 1860s Chinese gold rush shanty town.
Sad irony to my sparse recommendations is that like 7 years ago I directed a travel tv show - only tv show I ever directed - episodes on auckland, queenstown, wellington, and melbourne. And can't remember shit! Although to be fair a lot of what we did in the show was a bit $$$, scenic helicopter rides and wineries and whatnot. There's a lot of wine tasting goin' on in both Central Otago/QT and Auckland if that's your thing, but all that stuff bored me when we were filming it as I don't give a shit about wine!
