You would be much better off trying to get down in March. This is the start of Autumn, and the weather is generally pretty settled over most of the country. You could then start in the south and move north. However, given New Zealand's location, the weather is always going to be a little unpredictable!
Which island is best? Both, or should I say all of them. They are quite different, but where ever you go, you will find excellent photo opportunities.
Depending on airline and route, you can fly direct to Christchurch or Auckland, and if you go via Australia you can fly into Wellington (which is an experience in itself on a windy day!)
As for driving, if there is just the one of you, you could get a smaller camper van, which would get you to most places. For the more remote photo opportunities, you'd more than likely be looking at having to walk a little anyway due to the terrain. There are some places that rent 4WD camper vans, which could be a good compromise. Don't forget to ask beforehand if you want/need an automatic.
As has been mentioned, the driving distances may not be great by US standards, but there is little in the way of motorways/freeways outside of Auckland and Wellington. You won't want to cover large distances, as it can get pretty tiring. Long straight roads are the exception
If you like to walk / tramp, speak to some locals or DoC (Dept. of Conservation) first. Be prepared, the weather catches a lot of people out, and the facilities on all but a few tracks are pretty basic or non-existant. It is easy to get lost, even a relatively short distance from a road. There are no dangerous animals to worry about, apart from sandflies in the South Island
The North Island is warmer (generally) and more populated, but there is still plenty of space! Northland, the Coromandel and the East Cape areas are all nice, with beautiful coastlines and some historic towns. As you get to the middle of the North Island, you get the central volcanic plateau and the lakes. There are some fantastic areas here, and some great day walks. Have a look into some of the walks around the Tongariro National Park.
Over on the east coast, Napier is the Art Deco city. Flattened by an earthquake in the 1931, it was rebuilt in the Art Deco style, and has been well preserved ever since. Nice wines around this part!
The bottom of the North Island round Wellington is often overlooked, but there are some great photo ops, both natural and man made. The Wairarapa area is well known for its great wines, especially Pinot Noir. The south coast is pretty wild and rugged. Wellington has some great architecture, and the air is crystal clear.
There is plenty of info on the South Island here already, but one area many tourists miss is the north west part - Tasman Bay, Nelson, Nelson Lakes, and out to Farewell Spit. There are some stunning bush, mountain, lake and coastal scenes, some great wine and beer and very few people.
Another overlooked spot is Stewart Island, off the south coast of the South Island. If you decide to stick to the South Island, this may be worth a couple of days or so.
If you do come through Wellington, drop me a PM.