About a two hour drive northeast of Queenstown in the Mackenzie Basin, the small mountain community of Omarama is the type of place that is close to everything you've come to New Zealand to do (and even some things you wouldn't have thought to do). There are six lakes, three ski areas, and innumerable hikes within 100 km (about 60 miles) of this small town, making it not only a no-brainer destination from Queenstown, but one worth setting out for in itself.
The Ohau Snow Fields are the nearest skiing and snowboarding destination at about a 45 minute drive and are worth it for the drive in that opens up on Lake Ohau alone. Round Hill and Mount Dobson also have worthwhile powder, but are located further north near Lake Tekapo. The Ahuriri Conservation Park is just northwest of town and is a great space for hiking, mountain biking, and climbing some moderate peaks. You'll encounter everything from short and sweet routes from car parks to lake vistas to day long scrambles up some pretty technical tracks (i.e. Mount Huxley, the 2,505 meter/8,219 foot mountaineering feather-in-the-cap). There are also half a dozen mountain bike-accessible tracks, the majority of which are pretty easy.
The one thing we have failed to mention is the one activity Omarama is known for: gliding. Two schools operate in the area, providing glider (basically motor-less airplanes) flights of the spectacular ridges and valleys of this part of the world that include a handful of national parks. You can actually learn to fly one in a weekend if you're up to it! There are also plenty of fishing opportunities in the Ahuriri River if that is one of your interests. For more information on Omarama, browse the included links.
Note:One of the best things about Omarama is the number of stars that seem to linger above this little community; it's even better when they dim the public lighting during their winter stargazing festivities in August.