You can get in to Mt Cook by plane or helicopter, weather permitting of course.
As well as getting in and out, you can take a scenic flight, either in a helicopter or even a skiplane to the glacier. The skiplanes sometimes land on the glacier if the weather allows.
The Airport is situated right up the Mt Cook Road (Hwy 80), past the lake, by the river flats, and almost as far as The Hermitage.
Public transport to Mt. Cook Village is a bit tricky.
Only one of the big bus companies services Mt. Cook - this is Intercity/Newmans.
BUT: You can also use Kiwi Explorer, Magic Bus, etc. etc., get off at Twizel or Lake Tekapo, and from there travel to Mt. Cook Village by the local shuttle service, named Cook Connection.
The Cook Connection runs daily between Mt. Cook, Lake Tekapo and Twizel.
An additional service through the Waitaki Valley to Oamaru runs on Monday, Wednesday, Friday and Saturday only - but includes plenty of photo stops.
In Oamaru you can hop on the long distance buses to Dunedin or Christchurch along the coast, and in Twizel or Lake Tekapo the Cook Connection connects to the services to Christchurch and Wanaka/Queenstown.
Mount Cook is definitely a healthy drive from Christchurch, its nearest commercial airport. Since there were four of us traveling, we opted to rent a car, which turned out to be a very good move. Mount Cook likely has some public transportation options, but the car made it very convenient for us to get in and out of the park at our own schedule, for a decent price. On this particular occasion we went with Budget NZ, which provided us with a midsize car for a pretty inexpensive rate. Some point after this visit, we did get a flat tire, but the spare was in good condition and did not impede us from finishing the trip.
From Twizel a sealed road leads right into the park. Aoraki Mount Cook Village lies at the end of the road just below the terminals of the Hooker and Mueller Glaciers.
Aoraki Mt Cook village has an airport, but no serviced flights.
A regular bus service operates to Aoraki Mt Cook Village.
If going to Mt Cook by bus, do not forget to buy a round trip ticket as you may get stranded in this little town for longer than you had originally planned. With a bit of patience, you may however be able to hitchhike all the way to Christchurch (which I had to do during not-that-high season - November).
For more info and current schedules, you can visit my transportation tips page!