Steph said:
The Europeans are more oriented toward see that Africa or China, because of the shape of Europe (number of Peninsula, islands, length of coasts). So they were more likely to find America first.
America was an "almost" empty continent, with civs that were quite backward and relatively easy to conquer by Europe.
So the Europeans civs were able to get a large amount of new wealth at relatively low cost, and it boosted them greatly. Especially as they had to fiercily compete with each others.
You're on the right track but I think there's even more to say about this.
At the beginning of the 15th century, the Chinese Emperor Yongle launched several expeditions accross the oceans lead by the explorer Zheng He. During his multiple trips, the Chinese admiral went as far as Mozambique in Austral Africa and it is very likely that he actually even landed in Australia.
Finally, those expeditions lead to nothing, the successive Emperors being interested about the knowledge being brought back but considering there were no interests to settle in those empty lands which were too far away. Those expeditions were more about prestige than anything else, the purpose being to affirm the political leadership of the Imperial family.
In Europe, we all know things turned out differently, but the real question is why. Granted, great expeditions weren't meant strictly as prestige but had a commercial motive since the purpose was to find another way to bring in Europe Indian spices. But those spices were interesting only because of the political influence it could bring to the power controlling the route. If there were no competition between European powers to master those routes, there wouldn't have been such a rush.
The big difference between Europe and the other great civilizations is that Europe has never unified as did China, India, the Incas, the Aztecs or Arabia. Once all the reachable lands are controlled by a unique power, the political conflicts consist quickly in actually getting into power for the sake of power. On the other side, when there's no unification and a region is divided in multiple powers, the main political conflicts are about getting more influences than the other powers.
The reason why Europe hasn't unified is a matter of mere geography. The Roman Empire could reach such a unification because it was mastering the Mediterranean Sea. Considering that the Arabs were on the other side of the shore at the times of medieval Europe, such a unification was impossible because that sea couldn't be controlled. As a result, the only unification which may prevail would have been about continental Europe in itself., however, continental Europe is naturally divided by many barriers : Europe is divided in multiple peninsulas and the parts which are not peninsulas are split by mountain ranges.
That specific geography of Europe was not only making conquest harder, but it was also enhancing cultural differences, as physical geography was naturally putting Europeans apart of one another, those cultural differences making assimilation of conqueered lands even harder.
If Europe conqueered the world, it's simply because that civilization was divided in smaller entities being in competition against one another. The conquest of the Americas wouldn't have been so fast if there wasn't a competition between Portugal and Spain, with England, France and the Netherlands joining the game later.
As for the reason why Europe gets advanced faster than any other civilizations in military warfare, well, that comes from the same reason. As conquest didn't work because of the natural barriers dividing Europe, European rulers have constantly tried harder.