I have read that Columbus brought back 10 to 25 natives from his first voyage to the Americas. Seven or eight are said to have made it to Spain alive. Do we have any idea what happened to these seven or eight survivors?

When Columbus sailed through the Bahamas he took aboard seven Taínos. They would be brought to Spain with him, with the intention of teaching them Castilian and Christianity in order to aid with the conversion when they returned. These seven and a few others were then brought to the Castilian court in ~~1495~~ 1493, with the additional goal of serving as evidence of Columbus "discoveries". For Anthony Pagden they should also show the Catholic Monarchs that although the Carribbean proved poor in spices and gold, they might still be rich in "human merchandise" - Queen Isabella's attempts at breaking the Portuguese monopoly over the Atlantic slave trade had not worked out. But he also "brought them back as specimens, so that Their Majesties might see what people these Indies had in them". (p. 31)

[Edit: added paragraph] One of the first chroniclers of the Indies, Gonzalo Fernández de Oviedo is our main source for Columbus' arrival in Barcelona. There the native Americans where baptized. Their leader was baptized as don Fernando de Aragón, who was a relative of the important cacique or native leader Guacanagari (who had first welcomed Columbus on Hispaniola). Another, baptized after Columbus as Diego Colon, became an important interpreter for Columbus.

Oviedo then tells us in his Historia de las Indias

And another one they called don Juan de Castilla, and others they gave other names still, following their wishes [sic], or their patrons allowed to be given to them, in accordance with the Catholic Church. ... the prince [don Juan] wanted this [don Juan de Castilla] with him, and wanted him to stay in his royal house so that he would be well treaded as if he was the son of an important gentleman [or knight] whom he loved very much. ... and I [Oviedo] saw this indio who spoke already well Castilian, and after two years he died. All the other indios returned to this island in the second voyage of the Admiral [Columbus]. [my transl.]

So of the circa 7 Taínos all were brought back to the Carribbean to aid with conversion, except for one who stayed with prince don Juan until his death two years later. For Columbus there were no problems with taking these indigenous people captive, since at that point, they could still be seen in Europe as "barbarians" according to Aristotelian ideas - without having converted to Christianity they could be described as inferior, pagan and "less than human", and according to Columbus were "fit to be ordered about and made to work". The Spanish monarchs were thus very early trafficers in native slaves. While the Spanish Crown at this point started issuing decrees to protect the natives and to convert them, at first such commands were mostly ignored by Columbus in the Carribbean. Interestingly, Bartolomé de las Casas saw these Taínos in Seville as a young man. He would serve under Columbus and later become a strong advocate for the Americas' native population, which directly led to the Leyes Nuevas of 1542 officially ending native slavery (although it continued unofficially, on which more below).

These Taínos brought as slaves to Castille were a mere "footnote" for Columbus as proof for his own explorations - he loses interest when they are declared not to be slaves. But I think it's important to note for context that they were far from alone in their fate. First off: More slave shipments followed, including one of circa 600 Carribbean natives, and one of circa 500 Taínos in 1500 to Spain. Many of them died partly due to disease, but also probalby since they were completely uprooted from theid environments.

In 1508 a census listed that only 60.000 native people were left in Hispaniola (modern day Dom-Rep and Haiti) - there are estimates of ca. 3 million Tainos in the Carribbean before contact, although I'm not sure about the most current numbers here. Las Casas stated that by 1542 (the time of the Leyes Nuevas) there were only about 200 Taínos left in Hispaniola, a similiar fate shared by other native groups in the Carribbean. Charles C. Mann in 1493 notes that although no Taínos have survived today, according to modern research their DNA is possibly carried on by Dominicans of Afrcan or European descent today.

Second I'll briefly note that native slavery did not end abruptly with the Leyes Nuevas, and that this was a practice spanning the Spanish Americas, Carribbean, Portugal and Spain. Nancy van Deusen has written a great book ("Global Indios") on this, where she describes distinct phases: First between 1500-1542 "the enslavement of hundreds of thousands of people from America and elsewhere" (including Africa) due to the "open-ended exceptions of just war and ransom". Just war had served as a justification for war against muslims in medieval Iberia and continued to be used for conquest campaigns in the Americas.

A second phase begins with the New Laws of 1542 under Charles and heavily influenced by Bartolomé de las Casas. These already mentioned laws stated that native Americans were human, vassals of the Spanish Crown and free - effectively prohibiting enslavement of native people for just war or ransom. However, the New Laws included important loopholes which led to enslavement of natives continuing circa until the late 16th/early 17th century, albeit in much smaller numbers (numbering rather in the thousands regarding Castile). This meant that native people from Spanish America were still being brought to Spain at that time, often via Portugal. They would then use legal mechanisms open to them to argue for freedom, often succesfully. (I talk some more about native people and mestizos in Spain in this older post)

I'd like to close by steering a bit more away from the question: with some thoughts by Pagden on what effects such displacement could have on native Americans. He goes some more into the impossibilities for native Americans to be recognized or to express themselves. He's not talking here "only" about the Taínos brought over by Columbus, but also about native people brought to Europe by later expeditions, into the 18th century:

Samples of minerals and plants, once relocated in their new 'centres of calculations,' can be made intelligible by reference to other minerals and plants. Humans, however, rarely transport so well. They die or become meaningless in their new contexts. ... These creatures [sic] are only ever of interest as representatives of their own worlds. Although they are here among us, they are so only because we cannot easily go to them. If, like poor Pocahontas, they leave the shelter of their own exoticism, they very soon perish. ... Exposed to Europe, the 'savage' can only maintain his position of detachment for a short while. The critique of our ways, which nearly all such visitors express, will lose its force the longer the critic remains absent from the culture which made it possible at all. In Europe he runs the risk of becoming a mere ranter. What happens to him, as it must to all travellers, is a loss of identity. (p.31-32)

So that for Pagden, early modern Europeans tended to view native Americans in Europe as mere "curiosities" of an "exotic world". While for the native people themselves, their critique of European ways falls on deaf ears, and they lose something essential. Native slavery to Europe ended in the 17th century, but this did not make native people in Europe any more welcome or accepted.

Jon T. Davidann: Cross-Cultural Encounters in Modern World History
Anthony Pagden: European Encounters with the New World
Charles C. Mann: 1493. Uncovering the New World Columbus Created
William Marder: Indians in the Americas: The Untold Story
Nancy E. van Deusen: Global Indios: The Indigenous Struggle for Justice in Sixteenth-Century Spain
Gonzalo Fernández de Oviedo: Historia de las Indias

Edit: added a last part; plus a citation by Oviedo

Edit 2: added sources