Sounds alot like the Ardmore Flying Club which is just south-east of Auckland city (about 15-20 minutes from the International Airport). Its a good area of town (semi-rural) about 30-40 minutes from downtown in good traffic. Also at the same airport is the Auckland Aero Club.
There is also the North Shore Aero Club up on the northern end of the city which is a nicer area, still semi-rural but it is an upscale market - not exactly for the new, young student but more for the middle aged, mid-life crisis type.
Ardmore is the busier of the two, although both are on the outskirts of the Auckland Terminal Control Area, so getting up to do training can be tricky at certain points of the day (normally early morning and late evening)
Personally, I would suggest going a little further south-east to Tauranga. Bay Flight Centre has an excellent training centre, awesome weather and the control zone around Tauranga is almost custom built for training flights. I'd also suggest the Tauranga Aero Club although from what I hear they may not be doing so hot. Tauranga itself is like a small bit of paradise with The Mount for joggining, right on the ocean for surfing or swiming and a night life much like Aucklands.
I did all my training out of Rotorua, Whakatane and Tauranga so I know the area well, and its almost as perfect for Flight Training as you can get and with the exception of nor-easterly winds, the weather is almost perfect.