West Coast Region is long and thin, so it's an obvious choice to start at one end and finish at the other. It seems to work best if you do it north to south, since then you're not looking into the sun and the driver, at least, gets to see the sea views. The route here connects from Murchison in Tasman Region to Westport, Greymouth, Hokitika and Haast - and then pops out back into civilisation at Wanaka in Otago Region.
Further down from Haast is Fiordland. This has the best scenery of all (Doubtful Sound, Milford Sound, etc), but you can't just flip down the coast to get there: you have to go inland through Wanaka and Queenstown, and perhaps Te Anau, and then make the trip. See the Otago Region and Southland Region pages for them.
See the West Cost Region pages, or the links below, for more details of the places in the itinerary.
Here's the itinerary for West Coast Region, and as usual brackets () mean 'don't get your hopes up'.
Murchison
(Greymouth)
Hokitika Gorge
Wanaka