Neverwinter is correct about jetlag and driving. Have you booked your flights? If you have, are you able to change them? If I had only 10 nights, I wouldn't try to see bits of both islands, I would stick with one. However, a suggested altered itinerary:
Day 1 Arrive Auckland.
Pick up car from airport and drive to hotel. Check-in at hotel. Evening visit the Sky Tower observation deck
Day 2 Auckland harbour cruise at 1:30 pm. Remaining day at leisure. Have you considered doing a cruise to Waiheke Island instead?
Day 3 Auckland to Bay of Islands (Pahia)
> Drive up the east coast along the Kaipara Coast Highway.
> Stop at Matakohe and visit the Kauri and Pioneer Museum, have morning tea/early lunch at Matakohe House.
> The Waipoua Forest is beautiful, the forests of Waipoua are the garden of Tāne Mahuta - the "Lord of the Forest", the largest Kauri tree in New Zealand, with a height of 52 metres. it is said to be 1,200 years old. See photo In Māori mythology Tāne, the god of the forest, separated earth and sky and let light into the world. (There is much more to the story but that is the short version.)
> Stop at Opononi at Hokianga Harbour, there's a small visitor's information centre and hear the tale of Opo the gay dolphin, they show a black and white video from 1955.
> From Opononi, head east on the SH 12 (State Highway) go through Motutoa and Kaikohe, when you get to Pakaraka, turn left onto SH 10 and turn right at Puketona on Puketona Road, and stop at Haruru Falls. From there it's about 4 kms to Pahia. We left Auckland around 9.30am and arrived in Pahia just before 4.00pm.
Day 4 Hole in the Rock Cruise
> I wouldn't bother driving all the way to Cape Reinga, it's a long drive, you only have two nights and your time would be better spent in Pahia, there's plenty to see and do there. Unless the rules have changed since I was in NZ, rental vehicles are not permitted on 90 mile beach.
> Do a Hole in the Rock Cruise. See photo. They have morning and afternoon ones, I took a morning cruise, they stop at Urupukapuka Island in Otehei Bay photo here with time to grab a bite at the Zane Grey Cafe there. and they dropped me off at Russell on the way back at my request. I then caught the ferry back to Pahia.
> Go to Waitangi. If you are interested in Māori culture, this is a must. Some things to see -
> Ngātokimatawhaorua is a large waka (canoe) built by the Ngā Puhi people for the 1940 centenary of the signing of the Treaty of Waitangi. (It's believed to be the largest waka ever built.)
> Te Whare Runanga Marae. (The Marae, sacred open meeting area, generally situated in front of the "whare runanga", communal meeting house, is the area of greatest mana, the place of greatest spirituality ; the place that heightens people's dignity, and the place in which Māori customs are given ultimate expression.) The figure at the apex of the gable is Kupe, the Pacific explorer. There are a number of things to see in the Marae - totara (coronation chair), Wharenui, carved tukutuku panels and kowhaiwhai (scroll work on rafters).
> Waitangi Treaty House - this is where the Treaty of Waitangi (Māori: Te Tiriti o Waitangi) was signed on 6 February 1840 by representatives of the British Crown and a number of Māori chiefs.
Day 5 Bay of Islands to Queenstown
> Leave around 9.00/9.30 am. Stop at Whangerei for a coffee/break/ It's an hours drive from Paihia. The airport is a 2 hour 20 minute drive from Whangerei, depending on traffic. Drop off car, and get afternoon flight to Queenstown.
Day 6 Milford Sound day trip
Don't try and drive, it's a 5 to 6 hour drive each way, depending on traffic. Either do a tour or choose something else. The advice given to me was if you're going to go to Milford Sound and back in one day, keep the day before and the day after free. And this was if you're doing a tour.
> I would recommend BBQ Bus or Real Journeys . Real Journeys is a higher price and lunch is not included - that is extra.
> Intercity, New Zealand's national bus company, also offer tours. Their Milford Sound tour from Queenstown is from $119 (plus booking fee). If you are on a budget, this may be a better options for you. Here is their website Intercity
Day 7
Day at leisure. Evening visit the Onsen hot pool session. Currently we have taken the 9pm slot.
Day 8 Queenstown to Franz Josef
>After breakfast check out from hotel and drive to Franz Josef. Arrive Franz Josef in afternoon. Remaining day at leisure
> This is a long day, even without stops.
> Kawarau Bridge, we stopped here to watch people bungy jump from the bridge and see them swinging upside down over the river. Great entertainment. It's only 20 minutes from QT.
> Wanaka. Stuart Landsborough's Puzzling World is on the left. It's fun and some of it is free. Follow the road down to the lake. There's a large sculpture called "The hand that nurtures" - most people stand in front and get their photo taken. The lake is lovely, plenty of birds. Cafes etc.
> Makarora is half way between QT and Franz. Good place to have lunch. Scenery is great too.
> Lake Matheson. As you arrive at Fox Glacier, turn left at Cook Flat Rd then right at Lake Matheson Rd. From the car parking, it's a short walk around to the right to get to the lake. Great photo shot.
Day 9
Afternoon take the Glacier walk guided tour.
Day 10 Franz Josef to Christchurch
We stopped at these places:
> The Bushmans Centre is in Pukekura, rustic timber place with a cafe, shop and museum. When I was here, there were reindeer at the back and you could buy bags of "reindeer food" to feed them. Also, there were two quaint toilets in the air. It's a quirky and interesting place and I've not seen another like it.
> Hokitika - t's a coastal town and I walked to the beach - the sand is black. I've read recently that much of the sand is being washed away be erosion, so I don't know what it's like compared to when I was there.
> The train departs at 1.45pm and arrives in Christchurch just after 6pm.
Day 11 Depart Christchurch
> A few suggestions. You mention budget. You don't need a car in Queenstown, nor in Franz Josef. I would recommend you not hire a car, do a bus tour to Milford Sound and use the bus to travel to Franz and Greymouth. It will be much cheaper than car hire and petrol, plus you won't be stuck behind a wheel and will be able to look out the window at the wonderful scenery.
> The bus from QT to Franz is $62 for a non-refundable fare or $124 for flexible. (I typed in a random date). It leaves Queenstown at 8.10 and arrives in Franz at 4.10 with two meal break stops on the way.
> The bus from Franz to Greymouth departs at 9.15, arrives in Greymouth 1.15 with two meal breaks at the Bushmans Centre and Hokitika. Fare is $29 non-refundable or $57.
Your transport cost from Queenstown to Greymouth would be $91 plus the cost of whichever Milford Sound tour you chose.
> Instead of staying in hotels, stay in YHA hostels. I travelled around NZ for just over 5 weeks and stayed in YHA's and they're great. Not only is it far cheaper, you get to meet lots of other travellers too.
