The practicalities of the West Coast are fairly simple: most people drive the whole route themselves either in a car or a camper van, although I have seen many tour buses disgorging young backpackers at several stops on the route.
You can also use a public transport (train or bus from Christchurch to Greymouth, then bus) to travel along the west coast road, although the service beyond Fox Glacier towards Haast and Central Otago seems more erratic.
Surprisingly, we have not seen any hitch-hikers, at least going our way, but there is a fair amount of traffic and hitch-hiking should be easy, especially going north-south. There is a train from Christchurch to Greymouth which crosses the Southern Alps on a route sightly different from the road.
The distances involved are not particularly long (which is a nice thing about New Zealand: everywhere is not only scenic but quite near to everywhere else). It's about 600km from the Wanaka area to Christchurch via the Arthur's Pass, but Greymouth to Haast is only 314km (add 100km if you want to detour north to see the famous Pancake Rocks): easily driven in half a day, even factoring in a couple of short stops/walks, although to give the area any kind of justice you need at least a couple of days.
In fact, considering the weather, I would allocate more than that to the West Coast, to give it a chance for at least half day of not raining.
The optimal schedule would depend on the rest of your time in New Zealand as well as on your interests and manner of travel. If self-driving, I would suggest doing the route from the south to the north, which is what we did. We saw much more traffic the opposite way to the one we took, although obviously it could have been a coincidence. But even if it's not true, I think there are advantages here: you are likely to be less tired on the most interesting part of the route, and you are more likely to have some flexibility around the glaciers (and this is where you want it, for weather reasons if nothing else).
If you are not camping, Haast and Franz Josef are expensive accommodation wise, but their central location is such and advantage that you are bound to need a place in or near one of the accommodation: do your research and maybe even book before departure to get a chance of a better value place.
Fox and Franz Josef are also centres of all the "tourist activities", from normal walks to helicopter rides. There are many walks ranging from short half-hour loops to whole or even several-day hikes, all covered in local maps and leaflets available from tourist information centres. The only way to walk ON the actual glacier is with a guided group (think close to 100 NZD per 3 hour walk minimum), but you can walk to within 100m of the face for free.
Past Franz Josef there is more development, plenty of small resort or semi-resort type villages, but also less interest. I would suggest stopping somewhere between Hokitika and Greymouth and using a whole day (or more) for crossing the Southern Alps. Arthur's Pass has some fantastic walking options, and the Castle Hill area of limestone outcrops near Porter's Pass further on towards Christchurch on the same road is pretty amazing too. Lewis' Pass has Hamner Springs, which are great too.