The road from Queenstown to Glenorchy takes about an hour along a windy road which shadows Lake Wakatipu.
It's a very scenic drive but if you do happen to suffer from travel sickness I would recommend taking whatever precautions work for you.
For something a little different, you can catch the old steamship Earnslaw and sail up the lake to Glenorchy. It leaves from the Steamer Wharf in the centre of Queenstown, by the lake. This is a ship filled with history, and something from the past and our pioneering history.
In order to get to Glenorchy you do not have too many options. You can obviously drive your car through some absolutely astonishing landscape or hop into a bus. To my best knowledge, Backpacker Express, may be the only option. This small company's staff will pick you up at locations nearby your accomodation in Queenstown. They also provides good service to Routeburn, Greenstone, Caples, Rees and Dart Tracks.
If hiking the Routeburn track, at the end, you may want to get to Milford Sound or Te Anau with TrackNet. Just book ahead and they will pick you up! Alternatively, hitchhiking should also work.