I guess the
PR and Spanish systems are basically the same.
In Spain we go by our father and mother surnames. First the father then the mother. You usually skip the second surname and use it only when you need to clarify things.
Example:
I am Paco Anton, as you can see in my user name. My complete name is Paco (name) Anton (first surname, father's) Monllor (second surname, mother's). So I am Paco Anton or Paco Anton Monllor but never Paco Monllor or Paco A. Monllor.
My father is Paco Anton Lopez (his mother's surname). He goes by Paco Anton too and uses the full name when he needs to differentiate or to fill legal documents.
If I were to fill in US documents I'd go by Paco (name) Anton (surname) or Paco (name) Anton-Monllor (surname), most probbably the first as I rarely use the second surname and few people know it anyway.
Hope this helps