My mom slipped, fell and broke her hip in the Dominican Republic at a resort. She wants to sue but does not know where to start.

If the resort has contacts with New Jersey you can bring the litigation in New Jersey.  For example, if the resort was a Hyatt, or Sheraton, or some other type of resort that has locations all over the USA including NJ you can bring the suit here.  If not, it might require getting a lawyer in the Dominican Republic.  Good luck.