I would say that the first people in the region (if you are talking about the islands of the Caribbean) were the Arawaks. Although there is some controversy about this, most scholars believe that the Arawaks were there first and were then displaced by the more aggressive Carib people.
As far as the term "neoindian" goes, I really think this is just a semantic issue. There are those for whom the term is an insult because it seems to detract from the identities of these peoples -- it takes away the name that they give themselves and puts on them a name given by the white people.
So in terms of nomenclature, it is probably more accurate to call them Caribs and Arawaks. However, since these names and "neoindians" are just different names for the same people, it is not an issue of accuracy as to who was really there first.