What time of year are you travelling? Day length makes a big difference to how far you can travel in a day.
You simply cannot do both islands without flying. It might be best to spend 4-5 days in Queenstown and the remaining 4-5 days on a short road trip in the NI (Rotorua, Tongariro National Park, Waitomo Caves etc).
Or pick one island and do a more leisurely trip.
We fly to New Zealand 12/21/18, fly home 1/1/2019. Given the Christmas holiday I want to be sure we schedule now so as to get first choice at accommodations and adventures.
We fly from Fiji to Auckland on 12/21. Should we just transfer planes and continue on to Queenstown first, rent a car, and spend 5 days? Then fly back to Auckland and roam the north island for the remainder of the time? Thank you!
Flights from the USA generally arrive into Auckland . To reduce domestic travel you could overnight or two in the CBD to get over travel fatigue. Do a tour around Auckland sights, or day-trip to one of the Hauraki Gulf islands.
Hire a car and drive to either Mt Maunganui or Waitomo caves, then on to Rotorua. Spend 2 nights at least in Rotorua, and visit Taupo from this location. Fly to the South Island from Rotorua. Hobbiton can be visited from, or before, Rotorua.
Flying from Rotorua could be more expensive (check Air NZ's Grabaseat deals), but importantly you wont have that 3 hour drive back to Auckland airport.