If you do decide to take the ferry from the North Island to the South, plan on dropping your rental car before the ferry and picking up a new one on the South Island. Some rental agencies (Hertz for one) does not allow the cars to be taken between islands.
When you land at Picton, you could start your South Island. Isit with a 2-night stay at Lochmara Lodge, a nice eco-lodge accessed by water taxi from Picton. It is a very peaceful and relaxing place, with kayaks and SUPs available to guests for free. The property has a nice network of trails, and you can continue to the Queen Charlotte Track for more walking if you like. The restaurant served nice dinners when we were there.
https://www.lochmara.co.nz/
Upon returning to Picton, you can pick up a rental car there, or take the bus to Nelson and explore Abel Tasman National Park for a couple of days before picking up the car at Nelson Airport.
Things Tom see and do at Abel Tasman NP:
https://www.newzealand.com/us/feature/national-parks-abel-tasman/
We did a boat + hiking trip with Wilson’s, the major tour provider. They picked us up at our hotel in Nelson, and drove us to the boat dock, then returned us to Nelson by bus at the end of the day. Here are some of the activities and tours they offer:
https://www.abeltasman.co.nz/
Once you have the car, work your way south to Queenstown (allow at least a week for this, as there is lots to see and do on your way). Mountains, lakes, fjords, rugged beaches, rail-trails for cycling, whale watching boat trips, bungee-jumping, wine-tasting, botanical gardens, nice towns and little hamlets—-all on offer, depending on your interests.
The highlight of our New Zealand trip was a 5-day Milford Track inn-to-inn guided hiking trip, finishing with a boat on Milford Sound. You meet the group in Queenstown, and they take you to the boat dock for the track and return you to Queenstown at the end. You would return your car before the start, saving 5/days of rental fees.
https://www.ultimatehikes.co.nz/multi-day-guided-walks/milford-track
If this is above your budget, you can also do the trip unguided, staying in huts with shared sleeping space and shared kitchen facilities. But that means careful planning and carrying all your own food and other needs for the 3 days of walking. It is not for everyone, especially those without backpacking experience.
From Queenstown, you can fly back Auckland for your flight home.