That's a complicated question! Before I try and answer it, I would highly recommend, The Scramble for Africa, by Pakenham. An excellent book that covers most of the colonization of Africa in the late 19th and early 20th Century.

The short answer to your question is, it could vary. When you look at the nitty gritty of European colonization, it was often quite haphazard. The countries of Europe eyed each other warily and often made colonization moves out of fear that if they didn't, their rivals would. There also is a very common tension between groups w/in countries, with parties with business, religious, and/or humanitarian interests pushing for more control of foreign areas, and people (often the home government) who were against sinking time and money into these areas. Colonization was risky, it upset your neighbors, and very often did not result in a return on investment for the colonizing nation.

The method of colonization could also differ dramatically depending on which part of Africa it took place. Africa is/was a big diverse place. In some areas you had what we would consider true states (ex. Egypt), but in much of Africa things were more fluid and were built around kingdoms, clans, and villages. There was also an important divide in Africa between Muslims, "traditional" African religions (for lack of a better term), and African Christians. It's important to realize that before Europeans arrived on the scene in a colonial sense, these factions lived together and often came into conflict with each other (as did hundreds of other factions in Africa, based on tribe, kingdom, etc...). This dynamic provided both opportunities and dangers for Europeans getting involved in Africa. I could go more into the motivations of Europeans to colonize Africa (trade, religion, discovery, stamping out the slave trade, etc...), but I don't think that's exactly what you're asking.

So, from the perspective of an African (or African kingdom) on the ground, Christian missionaries were often the first Europeans in an area of Africa, but they were mostly at the mercy of the local rulers in terms of what they were allowed to do, and for their safety. But many were allowed to set up missions and proselytize, and this could create conflict not only between Europeans and Africans (Muslim and others), but between different sects of the Christian church as well. These in turn could influence decisions made at the state level in Europe when it came to Africa.

Business interests were also often first (or second) "on the ground." Legitimate and illegitimate businessmen looking to get rich in Africa. Africa had a reputation in Europe in the 19th Century as a "new eden," a place full of unexploited resources. These expeditions could range anywhere from a handful of people to large columns of armed men and porters (mostly African, but led and commissioned by whites) with supplies and items to trade. The main objective of these expeditions was to sign a treaty with the local ruler. Often for exclusive trade rights w/in the kingdom, excluding other European rivals. African rulers agreed to these treaties for a number of reasons. To get richer themselves, to get European aid against local adversaries, because they didn't know what they were signing, or because they felt they had no choice.

These relationships invariably led to "requests" for more concessions by the Europeans in the area. Inevitably conflict would develop between the Europeans and their hosts and the Europeans would often lobby their home government to intervene. Many times these efforts were resisted by the home governments, but they often eventually caved and sent troops in to tip the scales and get the local rulers back in line (or replaced). Many of these state interventions were rescue efforts where African kingdoms had had enough and were making war on the Europeans. These situations often captured the local European population's fancy, and the lobbying for the home government to do something could become enormous.

Once the state intervened, they rarely left. Often they found the local businessmen or charter company mismanaging things. Often the motivation for state intervention was not conquest as much as to stabilize a situation and to hopefully create an environment that did not require more state intervention. Colonization was often horrendously expensive.

Many times the "sunk cost fallacy" was at play. They had sent men and money to control a situation, so let's send in more to try and turn the place into a national asset (economically, strategically, etc...) that justifies the intervention. Then of course there was a prestige element that kept them going. Some areas of Africa, particularly in the South, became areas of white settlement, with Dutch, English, and Germans in particular sending thousands of whites to conquer and farm the land, or to extract natural resources.

This is a long answer, but it honestly barely scratches the surface. The motivations and method of European colonization in a place like, say, Egypt would be dramatically different than in the Belgian Congo or South Africa. But if I was to describe European colonization efforts in Africa in one word, that word would be "haphazard." The methods and motivations of colonization efforts by Europeans in Africa changed frequently and were as often reactive as they were strategic.