Cuba would be a pretty long trip....
Puerto Rico is more advanced than most people think. There is some pretty healthy flight training going on, so there are plenty of planes to rent.
So, the answer is yes. In San Juan there are a few flight schools at Isla Grande Airport (TJIG). Last I was there, there were two schools on the field and they rent aircraft. I believe you can take the planes to the USVI and all over Puerto Rico. For international flying you'd need to ask the specific school.
Enjoy your visit!
Checko