This was in 1492, when Christopher Columbus landed on what he called "San Salvador". While the exact location of this island is debated, most scholars believe it was one of the Bahamas. He then explored and claimed Hispaniola for Spain, establishing the first European settlement in the Americas.
From there, the Spanish went on to conquer many other Caribbean islands, including:
* Puerto Rico
* Cuba
* Jamaica
* Trinidad and Tobago
* The Bahamas
* The Virgin Islands
* And many others
So, while the first Caribbean island the Spanish landed on was Hispaniola, their influence spread widely throughout the region.