I'm quite certain that there are folks better-equipped to answer this here, but in the meantime I'll drop in with what I know :) My degree is in Near Eastern archaeology, but I've done archaeological work on WWI-era internment camps in Canada.
There isn't a particularly straightforward answer to this, because the Canadian west is so diverse in size and resources. First, I assume that you are referring specifically to European incursions west of Ontario, since aboriginal peoples had been "emigrating" place-to-place for thousands of years prior to European arrival.
As such, European trappers began moving into the west as early as the 17th century. This movement grew part-and-parcel with Arctic exploration, since Hudson Bay was a convenient landing place for expeditions into Manitoba, and the HBC held a trading monopoly on the watershed. The watershed was slowly explored in the interest of trade, and various permanent and semi-permanent trading posts, like Cumberland House were established.
From there, and with British Columbia being a bit of an exception, true settlement moved in a westward wave of sorts. Manitoba I believe is accredited as having the (highly debatable) "first European settlement in western Canada," on account of the Selkirk settlement beginning in the mid-19th century. These settlers were displaced by the Highland Clearances catalyzed by the Jacobite rebellion in the preceding century, and resulted in the enduring importance of various Scottish names, like Sutherland, in Manitoba.
For Saskatchewan and Alberta, the driving force behind settlement was Confederation. Where British Columbia joined the Confederation in 1871, there was a vast divide between the eastern settlements of Manitoba and the west coast. This led to a push for railway connection, which in turn necessitated settlements in Rupert's Land. It was known from surveying efforts for the railway that there was a significant amount of fertile land in between, so the Ministry of the Interior began immigration campaigns at the end of the 19th century and running up until the First World War to attract European immigrants from rural areas, largely Eastern Europe. The west was literally advertised as the land of milk and honey, and immigrants arrived in droves to take a chance at living in "the most progressive and prosperous colony."
