First of all, I assume that you are asking about immigrants to the area that is now the United States. If so, the answer to this question depends greatly on what you mean by “immigrants.”
The first people to come to the Americas were, as far as we know, from Siberia. They got to the Americas by walking across what is generally called a “land bridge.” In those days, much of the Earth’s water was locked up in ice caps, which is why the era was called an “Ice Age.” This meant that there was less liquid water and sea levels were lower. Because sea levels were lower, areas that are now shallow oceans were dry land. One example of this was the area that is now ocean between Siberia and Alaska. During the Ice Age, it was dry land and people migrated on foot, eventually crossing into what we now call the Americas. The gradually moved south and peopled what is now the United States. Their descendants are the people that we call the “Indians” or the Native Americans.
After these people came to the Americas, the next “immigrants” were Europeans. The first Europeans to settle permanently in what is now the United States were Spanish explorers who reached what is now Florida in 1565. They founded the settlement of St. Augustine, thus becoming the first European “immigrants” in what is now the United States. They came, of course, by ship.
In US History classes, we generally emphasize the coming of the English settlers since they created what is now our country. The first English “immigrants” in what is now the United States were people who founded the settlement of Jamestown in what is now Virginia. These people came by ship in 1607.
Depending on what you mean by the “first immigrants,” any of these three answers could be used.
The very first immigrants are most likely not documented, but I am assuming you mean the first United States immigrants. We learn that the first are from Europe, who started the first successful colony, Jamestown, in 1607. Before that, a couple Spanish immigrants from San Agustín (Saint Augustine) came to present-day Florida in 1565. I have not learned of anyone before that time in school, however, I'm sure a lot of "immigrants" went unnoticed/undocumented. No one can ever be sure who the first immigrants were.