The entire Caribbean is a huge question I'm not qualified to answer. I am more qualified to speak to your specific example, although I don't specialize in that topic.

Most indigenous Cubans (Taino, Siboney, and Guanahatabey) were the victims of cultural genocide, leaving little more than their names. In other parts of the Greater Antilles, there are contemporary 'Neo-Taino' movements, composed of people who claim varying degrees of indigenous ancestry.

If you were to perform DNA tests on the whole Cuban population, you would definitely get a lot of hits, especially in the Eastern provinces, but it isn't clear how many will be Taino and how many will be Yucatec. Between the 16th and 19th centuries, important numbers of indigenous peoples were brought to Cuba as indentured servants or, quite often, as slaves. This is why you'll see Cubans from the Eastern provinces (where many Yucatec indians were taken) with bronzed skin and other features (eye shape, hair, etc.) which look (and possibly are) indicative of indigenous ancestry.

The indigenous peoples of the island are largely from one of two origins. The Guanahatabey people of the Westernmost end of the island were likely from Florida, though this is still largely speculation as none have survived and archaeological evidence is not definitive. They had very limited tech and simple social organization, didn't really do much agriculture, and had yet to discover writing.

The Central and Eastern were populated by the Taino and Siboney people (sometimes collapsed into a single category of 'Taino'). They had island-hoped from the South, passing from the Amazon through what is now Venezuela and eventually arriving in Cuba. They had also yet to discover writing, but depending on where you're talking about they did achieve varying degrees of social and political organization (chiefs of chiefs) which went beyond simple familial connections. They also spun cotton clothing and knew basic agriculture.

The biggest hit to the Taino population was, as everywhere in the Americas, disease. This wiped out an enormous portion of their population.

Then came the fact that they were essentially enslaved by the Spanish. They were turned into encomendados (commendeds) through the encomienda system (commendation system). Basically, Spaniards would split up and distribute among themselves indigenous peoples to work for them, producing food and helping gather gold from streams (which was an important part of the colonial economy during its first decades of existence). The Spaniards, in exchange, were supposed to teach them how to be Christians. The indigenous peoples were being commended to European masters who would civilize and Christianize them, turning them into good, loyal Spanish subjects.

The result of the destruction of their entire way of life, brutal subjection, and the death of a huge part of their population, led many to suicide. Spaniards complained about how many were killing themselves or running off and fleeing to other islands. Some went into hiding in the mountains. In short, the population plummeted.

This led to a shortage of labor, which was 'resolved' by kidnapping more indigenous people from other places and brining them back to Cuba, establishing rules for how indigenous laborers were to be treated, importing slaves from Africa, and the implicit factor of market pressures. If the indigenous people were a commodity, their sudden scarcity upped their value and gave Europeans intensives to treat them more humanely, though not out of humanitarianism.

Many inter-bred with Spanish immigrants. Though records explicitly stating this are limited, we do know that most early immigrants were Spanish men, with a handful of women, and yet the population grew. It is likely that just as Spanish conquistadors married (or at least slept with) indigenous women on the continent, they also did so here.

Cuban historian Manuel Moreno Fraginals states in Cuba/España España/Cuba: Una Historia Común that local elites had policies of inter-marrying with Spanish elites from back in Europe and with bureaucrats and soldiers posted to Cuba, in an attempt to whiten themselves. Sources on this are paltry and I have yet to have a chance to read the perhaps 3 books which would give me better info on this point. One is Alejandro de la Fuente's Havana and the Atlantic in the Sixteenth Century, the pertinent volumes of Levi Marrero's Cuba: Economia y Sociedad, or a third book which I left in Cuba which studies the genealogies of the early elites of Havana. As you can tell, a lot of the research myopically focuses on Havana, but that's nothing new.

In any case, if you read authors like 19th century Spaniard Jacobo de la Pezuela (whose bias is that he wants to understand Cuba in order to make sure it remains a colony), you'll find smatterings of mentions of indigenous people in the 16th, 17th, and 18th centuries. They were always on the margins of society.

There is also a likely, though obviously difficult to document, relationship with runaway slaves and indigenous peoples already living in the hinterland. Once in the hinterland together they may have started families, but that's getting into speculation. Perhaps an expert on cimarrones (runaway slaves who would live in the Cuban hinterland) would be able to answer that more definitively.

By the 20th century, the Taino peoples of Cuba seem to have either died off or been culturally assimilated by the Spanish and African majorities of the island.