Columbus first made landfall on the island he called San Salvador (native name Guanahani), one of the Bahamas, and sailed through the Bahamas before reaching Cuba and Hispaniola on his first voyage. I'm not sure how many of the Bahamas he saw--perhaps Samuel E. Morison has the details in his writings.
The Dominican Republic has its modern name because Columbus chose the name Santo Domingo, after his father's patron saint.