Martinborough is the easiest wine growing region to get to. It's just over an hour's drive from Wellington on State Highway 2. Good pinot noirs are grown there, although Central Otago ones have more fame. Further north is the Hawkes Bay region (about 5 hours drive from Wellington).
We're most famous for our sauvignon blancs, and Marlborough is considered to be the best for this grape. Marlborough is a little more difficult to get to, being a 3.5 hour ferry. You can do day tours or you could spend a night in this area, or extend it to 2 or more to visit Nelson and its wines.
The New World supermarkets in Wellington CBD have very good wine and beer selections, and there are some specialist liquor stores about. Regional Wines and Spirits is on Waring Taylor St (off Lambton Quay) and there is a wine shop on the opposite corner to Wellington's i-site (other side of the traffic lights).