This is a very common question for whoever wants to visit the South Island from Auckland. Here is a brief summary of the two options:
Flying to the South Island
Flying is definitely the cheapest and fastest way to get to the South Island BUT when you get there you will need to hire a car. Public transport in the South Island is not so developed and you may not be able to reach all places.
Driving to the South Island
It’s not a short drive but if you don’t mind, it may worth it. It takes roughly 8hrs from Auckland to Wellington, then 3 and a half hours on the ferry. You will arrive in Picton and, from there, it’s another 10 hours if you want to go to Queenstown. That’s why we only recommend you driving if you plan to visit the North of the South Island.
If you fly and you need to hire a car, you may well end up spending the same as driving down. If your plan is to hire a van, remember that is not cheap and you may end up spending less staying in cheap hotels (although the experience of driving around in a van in the South Island is priceless).