The picture I have in my head is loading people, animals, and food onto ships, and saying "Good luck!"

In some cases, you might not be far off.

What would a colonization attempt have looked like?

The nature of the colonies was largely driven by the circumstances by which they were created. Unfortunately, for most of the colonies in the pre-Roman period there is little surviving primary evidence, however, we can generally differentiate these colonies (largely Greek and Phoenician) as one of two types. The first, and by far the most common, was little more than a trading post. We can find parallels in the early colonies along the North Atlantic during Europe's age of exploration. Examples of this include Cathage, Naples and Masillia. These trading posts might grow into larger settlements as population pressures from the mother city drove people there, or, as in the case of Masillia, the mother city being destroyed. The second case is purpose built colonies. Tarentum is one such example. Founded by Sparta, Herodotus reports that it was initially colonized by what amounts to bastard sons. The population pressures on the society of the polis drew many people to the new opportunities as land and thus wealth became increasingly hard to come by. We can assume that those pressures are what drove many to the new colonies, whether purposefully set up, or mere trading posts.

Were promising locations scouted out first?

We can assume so yes, in most cases, although I can't think of a source about ancient Greek or Phoenecian colonies that explicitly states it. Rather, it seems to be more reports about an anxiety that finding locations for new colonies was difficult. Whether the new colonists had any mind of what their new surrounds would look like is another story.

Why were they sent out in the first place? How were colonists selected for the expedition?

I wanted to expand on this in relation to my answer above. So, I said 'population pressure' and that is a general description for what amounts to a complex problem. The stability of most ancient cities was always razor thin. The Sparta example is a good one, because even if it is untrue it illustrates the idea nicely. In Tarentum's case, these men were basically not citizens, but possibly could claim citizenship, by force. Sending them off to Italy solved the problem for both parties (although clearly more in the favour of the citizens of Sparta than those being sent away). If they weren't sent away, they may have completely upset the balance in the state. We have to remember that the resources for any one particular city were almost always very limited, and natural population pressures meant that the only options were conquest, colonization, or collapse. Those selected for the colonies meant as actual settlements more than trading posts are almost always associated with exiles, indicating the possibility of a political focus for their selection. However, common sense would lead us to believe that it would be those most likely to upset the existing balance that would be selected (and also be willing to go).

However, as I stated above, for the pre-Roman period, a lot of this is guess work. Most of the important colonies were founded in the 8th and 7th centuries BCE, and their founding myths can't be taken strictly at face value. The Roman period is very different though, because their colonies were not merely driven by population pressures at home. Sure, the selection process was driven by similar criteria (the landless, former soldiers; basically people who could cause trouble). However, there was also the intended imposition of power on conquered peoples. Colonies of Romans were settled everywhere, first in Italy during the conquest of the peninsula, and then all over the future Empire, particularly in the West where existing cities were few and far between. Many of the colonies grew into large and important cities, and several modern cities can trace their lineage back to them. Many did not. However, I think that the design of the average Roman colony speaks volumes about its ultimate purpose: square, grid lines, laid out like a military camp. And like a legionary camp, they were designed to impose Roman order.