I spent a little while staying with family in Auckland, so I hope I can address some of your concerns.
Some of the most popular suburbs in Auckland include Ponsonby, Parnell, New Market, and Mount Eden. I spent a fair amount of time in the North Shore area, and there were quite a few suburbs there that seemed really nice, with a range of accommodation to satisfy a variety of budgets.
I know that there is a substantial Indian population in West Auckland, specifically the Waitakere area, if being in a culturally familiar environment is important to you.
A good resource for finding accommodation in New Zealand is TradeMe, the local equivalent of sites such as Craigslist.
As for your children, bullying is unfortunately a reality as it is in many schools, and there is the odd racially motivated incident. As much as bullying and racism should always be condemned, these factors have not stopped an expanding Asian population from settling there. You can have a look at our Education and Schools in New Zealand page to get a better idea of the system there.
I hope that you have a successful, happy move.
Best of luck,