Hi Bindu, and welcome to the forums!
If you haven't already, take a look at the NZ Starter Kit
- there may be something in there that will help with your plans.
First off: you won't have any problems driving around both islands. You may
be required to fit snow chains in some parts of the south island (particularly the alpine ares) but you'll get plenty of opportunity to find this out beforehand. Certainly don't worry about it.
Both of the options you've stated will work, and both will give you the scenery and experience you're looking for. The problem (for want of a better word) with New Zealand is that it's all
impressive, so you won't really be disappointed either way. Plus, each island will give you a whole bundle of contrast: the north will give you rolling hills, pristine beaches, heaps of culture and thermal activity, whereas the south will give you snow-capped mountains, mirror lakes, glaciers, valleys and raging rivers - and of course the Fiordland!
I guess it's really down to what you want to see the most of, and as you've only got nine days it's a toughie. Have a think, but bear in mind that if you do want to see the Fiordland and the lakes and mountains, you'll need to get as far down as the Southland. This will mean heaps of driving (especially as you're starting in Auckland) and you may want to consider that flight in between to alleviate that. There are plenty of hire companies that will let you collect a different car after the flight and continue with your journey, so you'll get around it.
I'd happily suggest some alternative routes but it really depends on what you want out of the trip. What are you interested in most? What are your 'must-do's'?
If you can give us some ideas we should be able to help you come up with something that'll work..