Mount Cook Village is in the central South Island, quite close to the West Coast but separated by mountains. It's a 200-mile (330-km) journey from Christchurch by road.
By Private Car
Duration: 4 hours
Many travelers to New Zealand like the freedom of renting a car. When driving to Mount Cook Village from Christchurch, take State Highway (SH) 1 south-west and keep going until you see signs for Geraldine/Aoraki/Mount Cook. Then, turn onto SH 79. The road west turns into SH 8 at Fairlie. Continue along this road past Tekapo, then turn up onto SH 80 from Pukaki, following the western shores of Lake Pukaki. The roads are all well sign-posted.
The first part of this journey, across the Canterbury Plains, is quite flat. After Geraldine, it gets more hilly, and the views more attractive. You'll probably want to stop to take some photos as you near Lake Tekapo and Lake Pukaki.
Duration: 4.5 hours
There's a direct bus between Christchurch and Mount Cook Village if you'd rather not drive yourself. It follows the same route as mentioned above. It's a bit slower than traveling by private car, but you'll get to enjoy the views without having to worry about driving. Buses are reasonably comfortable and spacious, and they do stop for toilet and meal breaks.