Chinese junks, similar to what Zheng He's fleet may have looked like. Credit: Wikimedia
Exploration is rarely about the pure, unfettered pursuit of knowledge. Scientists want to prove a theory. Politicians want to see high-tech industry stimulated and their national prestige elevated. Private citizens want to go "because it's there." History has shown that there are two broad kinds of exploration - with their own separate outcomes. The first is the show of force. The second is sustained interest.
1405 saw the launch of Zheng He's expedition of exploration from China. A massive flotilla of 317 ships and 27 000 crewmembers, with 44 huge treasure ships measuring 120-150 metres in length. Or, accounting for typical historical exaggeration, probably half that as 100 metres is the limits of what is possible with wooden ships (note that no remains of these craft have ever been found).
Christopher Columbus had 3 ships, 23 metres long, and 270 men (before the usual diseases started wiping them out). The only thing he had over Zhang He was the fact that he didn't sing soprano (Zhang He was a eunuch).
The Chinese weren't interested in colonisation or trade. They wanted to show off, impress people with their bling (hence the treasure ships) or else kick them around a bit. They wanted to remind people that China, just like every other Empire in the world, was the centre of the world. It was the Apollo project of the era, in more ways than one. Because after a flurry of these expeditions, the whole thing was called off due to escalating military conflicts and finally a big damn wall to keep the Mongols out.
Christopher Columbus of course, had different motivations. He wanted money by opening up an alternate trade route to Asia. And he didn't make such a big investment - three ships, perhaps a big thing in Medieval Europe. Although everybody thinks of him as a big success because he found this place called America, he failed to find the trade route. He wasn't even out to prove the world was round, everybody knew that.
Now, how did the New World wind up colonised? Why didn't the Chinese do it? Europe was a dirty little backwater, although the first glimmerings of scientific enquiry were beginning. Simple: China wasn't interested. China could have easily afforded to colonise and conquer these new lands. The Europeans were broke, and constantly fighting each other. But there was land to exploit, and savages to convert! There was also somewhere to run away to if you didn't fancy being oppressed!
So, we have two scenarios with which to place interstellar colonisation in. The first is the massive show of force, done mainly to impress and sustained by government interest. The second is a trickle which eventually becomes a flood, sustained by continued private interest (Hey Sven! Could I interest you in a place called Greenland!). This can tell us something about what kind of colonies we can expect, and who's going to found them.