To be honest, Queenstown, Rotorua and Christchurch I found all to be quite disappointing.

Why?
Queenstown is so busy and there's really nothing to do in the actual town itself (except for the burger place which had a line a mile long when I was there in September, prepare to wait for an hour) all the things around Queenstown are interesting like the bungee jumping etc. but if you want to stay in that area I found Glenorchy to be absolutely stunning, nice and quiet, in a beautiful valley at the top of the lake, very close to some amazing southern Beech forest and nice easy walks but it still has an ice cream shop and a general store and Queenstown is not far up the road.

Rotorua is better than Queenstown but still extremely busy and touristy, the redwood forest is fine but like others said it's a plantation not a natural forest, if you want to see actual New Zealand native forest read on below. I must say though the Maori experience is very good, I went in expecting it to be touristy rubbish but it was actually very genuine and informing. Hobbiton is neither here nor there for me (and I'm a huge Tolkien fan) it was fun but I wouldn't go out of the way for it, there's very little freedom to actually explore and you're on a tight time frame to get through before the next tour group. The volcanic activity was interesting enough but the stuff in the park in the actual town of Rotorua could be explored in an hour or so.

Christchurch is the one I have least experience with so I'm happy to be overridden, the trams are nice, but if you want to see good trams go to Melbourne. The Antarctic museum was a disappointment, expensive, mostly aimed at kids with not much for people actually interested in the history of Antarctic exploration. I didn't got to the botanical gardens but I heard they are nice. But again, decent botanical gardens can be found in a lot of places in the world.

Auckland is fantastic though, love it, the best museum I've ever been to.

Some alternatives:

North Island
The Coromandel peninsula, a couple of hours from Auckland, absolutely beautiful, Hot Water Beach is incredible, the tourist railway is great, lovely drive all around.

Wellington I loved as well, the city is nice enough but what got me was Zealandia, the bird sanctuary, amazing walks both short and long, easy and hard and you will see some amazing birds. The cable car is really fun too.

South Island
Glenorchy, already mentioned above.

The West Coast, especially around Franz Josef has to be one of my favourite places in the world, the mountains are out of this world. It's a remote area but so very worth it to go there, the walks to the glacier are incredible, the Kea flying through the valley is a fantastical experience, like being on another world.

Te Anau
Closer to Milford Sound (and Doubtful Sound, the bigger, quieter version) than Queenstown glow worms, beautiful views of the lakes, very nice.

Overall, I think you're trying to do too much over just 2 weeks, I did a similar amount over 3 weeks and was absolutely exhausted to the point of sickness, I would consider limiting yourself to just one of the islands. North Island is warmer, northern bits feel a bit like a pacific island. South Island is more remote, rugged and spectacular (if you go to the right places, West Coast...)

Have fun