This is an extremely USA-centric question. I really can't be angry; world history education in some parts of the US is terrible or non-existent. I strongly recommend reading a good basic world history, like the one Robert Strayer has written (see below).

In the meantime, the basics: Christopher Columbus was from Genoa (an Italian city) who sailed for the king and queen of Spain. The Spanish started establishing colonies right away after Columbus found this "new" land. First they took over some Caribbean islands, then turned toward the mainland (though they weren't quite sure it was the mainland for a while). By 1520 they had conquered the Aztec empire in the Valley of Mexico and were gunning for the even larger empire in the Andes mountains of South America -- the Inca (or Inka) empire. They set these up as colonies and started expanding beyond them.

The Portuguese were exploring the west coast of Africa throughout the 1400s, and got into the Indian Ocean in 1498. This is why the Spanish monarchs (technically, of Castile and Aragon -- there was no "Spain" yet) were willing to take a gamble with this nutty Genoan who said sail west instead of south and east to get to India. The Portuguese touched down on the eastern tip of South America in 1504, and they, too, started colonizing (this became Brazil), along with establishing bases along the African coasts and in the Indian Ocean.

The Dutch, French, and English started competing (weakly) with the Spanish and Portuguese in the late 1500s, all three exploring, and establishing colonies in, North America in the late 1500s and early 1600s. Spain and Portugal weakened as global powers in the 1600s, replaced by the Dutch, who founded their North American colony, New Amsterdam, in 1624. In the meantime, the English colony of Roanoke, founded in the late 1500s in what is now North Carolina, collapsed and vanished. The colony at Jamestown in what is now Virginia, established 1607, as you said, succeeded, though barely.

So Europeans started colonizing right away. The English, in particular, started seriously colonizing only in 1607.

What I've written is a painfully bare bones account of a rich, fascinating, horrifying, and complex story. In the meantime, Robert Strayer, Ways of the World, whatever the latest edition is. That's a start. There's WAY more where that came from.

Colonisation was prompt. Depending on what you call a "colony", you could even say that colonisation began within a year, with the construction of Puerto de la Navidad (in what is now Haiti) by Columbus at the end of 1492, with a fortress and small settlement (of some 3 dozen men) built from the wreckage of Santa Maria. If the settlement had survived until Columbus's return just short of a year later, it might well have become the first permanent European settlement in America. As is, all that remained when Columbus returned were charred remains. On his return, Columbus came with 17 ships and over 1000 people, including women, and crop plants and livestock in order to establish a substantial colony (this fleet was about the same size as the British "First Fleet" which established the first European settlement in Australia). Columbus's next settlement, La Isabela (Dominican Republic) was built, and had over 200 buildings including a church.

La Isabela doesn't appear to have been well-located, and the settlement was abandoned in 1496, and a new settlement, Santo Domingo (Dominican Republic) was established by Bartholomew Columbus (Christopher's brother). Santo Domingo is still inhabited, currently a city of about 3 million people, and the capital of the Dominican Republic. (The settlement was moved to the other side of the river, from the east bank to the west bank, by Nicolás de Ovando, the third Spanish governor of their American colony, on his arrival in 1502 (with a fleet of 30 ships and about 2,500 colonists).

La Isabela was a definite colonisation attempt, and Santo Doningo a successful one, all within a few years of Columbus's first voyage to America. Spanish colonisation in America grew enormously in the next few decades with their conquest of Mexico in 1521 and their conquest of Peru in 1535. 1565 saw the first permanent European settlement in what is now the USA, when the Spanish established San Agustín (now St. Augustine) in Florida. The Portuguese sought to establish colonies in Brazil from 1530, and if one is happy to call a trading post a "colony", then the French also tried to colonise Brazil at the same time.

By the end of the 16th century, over 300,000 people had arrived in the Americas on European ships (not all voluntarily - this total included about 75,000 slaves from Africa). That Britain, France, Sweden, and the Netherlands only successfully established their colonies in the 17th century just makes them latecomers compared to Spain. (Britain did try to establish colonies starting in 1585; Jamestown in Virginia in 1607 wasn't their first attempt, just their first successful attempt).