Spain had a leg up on the others since they were willing to fund an adventure that bucked the going dogma of the day, that the earth was flat. Of course, Christopher thought he was actually going to Japan, known as the "Far East", by going West! Imagine that, heading the opposite direction you wanted to go and expecting to end up there. The French, Dutch, English, Swedish, and Russians certainly did not want to be left out of the colonizing business and started their own explorations using their own ships.
The French managed to slide down the St. Lawrence and name a bunch of settlements soon to be called Canada. They also managed to slip by the Spanish and settle a place at the mouth of the Mississippi called "New Orleans" after the "Old Orleans". The Dutch and Swedish focused their settlements along the upper coast of North America called "New Amsterdam" after the "Old Amsterdam". The English, fighting the Spanish all the way, split the difference between St. Augustine to the south, and New Amsterdam in the north, calling their new land "Virginia" after their unmarried, virgin queen, Elizabeth. Of course the Russians went down the other side of the Continent that didn't exist before Christopher stumbled into it, opening the doors to Alaska and the upper Pacific coast.
Besides "Gold" and "Glory", a major factor in driving this arms race, was "God"! Each country had settled into a religious warfare that sought to control the hearts and minds of those participating in this colony-building enterprise. Spain had become the defender of the Catholic faith and was rewarded by Pope Alexander VI who issued two international decrees (Papal Bulls) that all land west of a line running north-south through the Azores and Cape Verde islands would belong to Spain. [Somewhere between the 41st and 44th meridians west of Greenwich.] The land yet to be discovered lying east of this line would, of course, belong to another Catholic country, Portugal. These decrees were issued the 3rd and 4th of May, 1493, at request of Ferdinand and Isabella of Spain.
The Protestants (English, Dutch, and some French) were left out in the cold, literally, since their only options were to explore northeast or northwest, freezing in the Arctic circle. Priests, ministers, pastors, elders, and all sorts of church leaders were involved in the planning and practice of this nation-building enterprise. This religious fervor would be a major driving force to many who were yet to come to these new lands. These forces continued to thrive even to the very building of our little town, Winchester, KY., Pop. 8,000.