When Columbus arrived, the majority of Cuba's inhabitants were Western Taino, closely related to the people of the Bahamas and Jamaica. They called themselves the Ciboney, but this term has been confused with their Guanajatabey neighbors in westernmost Cuba, so you'll sometimes see the Guanajatabey erroneously referred to as Ciboney instead. This map gives you a basic break down of the area.

The Ciboney Taino came from South America. To be a bit more specific, they're generally thought to have come from the Orinoco River basin in Venezuela. The evidence for this includes linguistics (the Taino language is part of the larger Arawakan family, which is widespread through portions of South America), similarities in pottery styles, and similarities in agricultural systems.

The Guanajatabey, however, appear to have been a non-Taino population who spoke a very different language and did not practice agriculture. We don't know much about them today. They're thought to be the remnants of earlier inhabitants of the islands that came from Yucatan (or a little less likely, Florida) perhaps as early as 5000 BCE. The Taino themselves began settling the Caribbean around the 2nd Century CE (if not a bit earlier). Following the Taino there were other influxes of South American populations, representing the Igneri and the Island Carib.

As for how the Taino and others got to the Caribbean, they employed canoes. In fact, English gets the word 'canoe,' ultimately, from the Taino. Taino canoes were long, narrow dug-outs, famed from their speed, maneuverability, and for being virtually unsinkable. When Columbus first arrived in the Caribbean, one of the first vessels he encountered was a Taino canoe that was as long and held as many men as his own flagship. Among the Island Carib, some of their canoes had been made by the Warao on the South American coast and traded at least as far north as Dominica.