This will deal specifically with the impetus for Britain's Canadian colonies to join Confederation as the Dominion of Canada in 1867, as the entire history of Canada from the Age of Discovery to 1867 covers more than four centuries and is too broad a topic for just one thread.

You may find some answers to your question in this thread by u/enygma9753 about the post-US Civil War era, Fenian raids into Canada and the pressures this placed on Canada to unite as a country.

France ceded its control over Canada to Great Britain after the Seven Years War (1756-1763). Britain had captured Quebec in 1759, ending the 150-year-old New France era in Canada. It became a British colony in 1763. Over time, French Canadians became British subjects with unique rights that protected their culture, language, Catholic faith and property. They had little connection to continental France after the 1759 Conquest and resented the fact that France had willingly surrendered New France at the peace negotiations, viewing it as an abandonment or even a betrayal by the mother country. (France had chosen to keep Guadaloupe, a prosperous sugar colony, over less profitable Canada.)

There were fears that Britain, who still had responsibility for Canada's defence in the years following the 1867 Confeferation, may not be as interested in providing for Canada's security in the future and this issue came to a head at the end of the US Civil War, which had much more impact on the creation of Canada as a nation than France's brief imperial foray into Mexico.

Both Canada and Britain had real concerns that a victorious Union army could invade Canada and annex the poorly defended colonies. The cross-border Fenian raids in the 1860's (discussed in the link above) fueled such worries. The Canadian colonies -- modern-day Ontario, Quebec, New Brunswick and Nova Scotia -- felt that uniting as a country would be the best way to ensure their collective defence and economic interests. Canada acquired the bulk of its western territories in 1870 when it purchased the Hudson Bay Company's vast land holdings in Rupert's Land and the Northwest Territories in 1870 for $11 million dollars.

Canada also made a series of unequal treaties with the indigenous peoples to obtain control over their lands, in both colonial and post-Confederation periods. When Canada added the province of British Columbia on the Pacific coast in 1871, the government didn't recognize indigenous title to lands there, didn't feel obliged to even make treaties with the region's native peoples and simply acquired the territory outright.