What nation first established colonies in the New World?

Spain.

More Information
The first European settlement in the New World was Vinland in (what is now) the Canadian province of Newfoundland and Labrador around 1000 C.E. However, it was destroyed by the First Nations who lived there.

The first European colony in the New World was Hispaniola in (what is now) the Dominican Republic and was established in 1493. The colony of Hispaniola was founded by Spain.