8 days in New Zealand?

Since Auckland and Queenstown are separated by about 1,000 miles, most visitors choose either the North Island or the South Island for a short visit such as you’re planning.
You could fly between the two cities and rent a car to drive down to Milford Sound from Queenstown. That’s a drive of about 4 hours one way. And driving in New Zealand is different than in the U.S.—often with one-lane highways, winding roads and one-way bridges.
You might opt to visit just South Island for the 8 days you have. If you fly into Christchurch, you could go to Lakes Tekapo & Pukaki, Fjordland National Park (Te Anau) Doubtful Sound, Milford Sound and Queenstown.
With more time, you could continue up the west coast to Punakaki and Kaiteriteri, where Abel Tasman National Park is located.

Auckland was fine, but not the most interesting spot we visited in New Zealand, by far. I would recommend following the New Zealand Travel Tips, Facebook page. You will find knowledgeable locals that will provide invaluable information. The group was started by a gal named, Sarah Chant. She built the group and there is a corresponding website. There are many discounts through NZTT. I know Facebook has its issues, but totally worth being on Facebook just to follow this group. NZTT. I have seen many first time travelers get into some kind of bind in New Zealand and either Sarah or other members are quite helpful in the resolution. We did both the Milford and Doubtful Sound boat trips. AMAZING! You could also search this forum for previous posts.

With 8 days, you really need to pick one island or the other, or spend a lot of time driving without stopping.

Just a quick gauge on travel times.
Auckland to Wellington is about 8 hrs.
Another 4 to 5 hrs from Wellington across Cook Strait to Picton.
11 hrs Picton to Queenstown.
4 hrs Queenstown to Milford Sound.
Typically a standard Milford cruise is about 90 mins.
A flight from Auckland to Queenstown is around 2 hrs in length.
I would recommend travelling from Queenstown to Te Anau (a pleasant 2 hr drive) and to start and end your Milford Sound journey from there rather than from Queenstown. I also recommend booking a guided coach and cruise tour rather than driving yourself. Typically a tour from Te Anau will include a 3 hour drive along Milford Rd (stunning) with a number of stops. Then around 90 mins of cruising on the Sound.(actually a fjord). Then about 1 hr 45 m back on the Milford Road without the stops. There are a number of other options but that is the basic and very affordable option.
From Te Anau a guided coach, cruise tour of the larger more remote Doubtful Sound can also be done.

Can you do what you are looking at doing in 8 days? Yes you can. Would I recommend it, not in a million years but each to their own.
You will need to coordinate a domestic flight from Auckland to Queenstown.
From Queenstown you can fly to and from Milford Sound.
You could then fly from Queenstown back to Auckland.
So in theory you can do it all in 4 days or maybe less.

Both Air New Zealand and Jetstar (Qantas budget affiliate) operate frequent direct flights between Auckland and Queenstown, so yes you can conceivably base yourselves in both locations for the time you have available ... but anything further afield will be hard to fit in.
The country is tailor-made for explorations by car - particularly the South Island countryside around Queenstown. Day trip excursions to Milford Sound, Te Anau, or nearby Arrowtown would make for fast-paced itinerary , but if that's your mode of travel it would be doable.
In your case the trip to Milford Sound would probably be best done via a local tour from Queenstown - maybe as an overnight trip given the long distance involved.

We stayed in Te Anau 4 nights and drove to both our Doubtful Sound and Milford Sound cruises. Highly recommend. You can stop where you want to go at your own pace. Te Anau is a charming town and we enjoyed a couple of restaurants and did some hiking.

Here are a few well located hotels in Auckland. You want to be near the waterfront/ Viaduct / Britomart.

M Social
QT Viaduct Harbour
Movenpick

In Auckland I’d recommend a guided tour of Auckland museum on your first day to get your bearings on the country. Check their website for tour times and special events on the What’s On page - https://www.aucklandmuseum.com/whats-on

A day trip to Waiheke Island from downtown Auckland is also a great way to spend a relaxed day. Have a look at tours with Ananda or others on the island.

For your Milford Sound visit I would recommend guided tour bus to get there, cruise on the sound (fiord actually…) and fly back to Queenstown. It’s pricey but you won’t regret it. Here’s one example (there are others): https://www.altitudetours.co.nz/tour/milford-sound-coach-cruise-fly/

Colin has good advice that answers your question.

I can’t answer it because with 5 weeks in NZ this year, I spent no time at all in Auckland (except the airport). I think it’s a nice city but for me, the highlight of NZ is the scenery. I am also an experienced traveler in my 60’s and had a car for most of my trip. With your priorities, though, a car might not be of much use.

There’s a lot to see on the NI, but with only 8 days I would be advising friends to prioritize the SI, utilizing flights. Head straight on to Queenstown when you arrive and then Milford Sound however you decide. From there, do you want nature or city? I quite liked Christchurch, so I would probably start in Queenstown and end in Christchurch, flying back to Auckland from there for your last night before flying out.

Don't miss Oamaru to see the penguins, Akaroa to see the Hector dolphins, or Abel Tasman National Park.

end in Christchurch, flying back to Auckland from there for your last night before flying out.

Pointing out that there are scenarios without in and out of Auckland since United flies SFO>>CHC nonstop.

So using a Star Alliance ticket this is possible on one ticket:

Minneapolis>>Houston>>Auckland>>Queenstown, a mix of United and Air NZ, a very long day (I’ve done this trip).

Then return Christchurch>>San Francisco>>Minneapolis all United.

The temptation from MSP is to use Delta which does codeshare with Jetstar so that could work, but there are no 2 flight options to the South Island with Delta.

Responding to the initial post prioritizing Auckland (but I wouldn’t): Fly all the way to Queenstown the first day. Spent the bulk of the nights around here, fly back to AKL, then spend 3 nights there before returning home.

Great info for flying directly to the South Island from Minnesota. We had 5 weeks, so we covered both islands and used the ferry between the two. We were lucky, the weather was beautiful when we used the ferry. But, the ferry takes about four hours, if I recall correctly, and the ferries are cancelled not infrequently due to weather. When we go back we'll be flying between the islands.