What is the best way to get started in this dive?
First, you need to decide if it’s something you really want to do.  Do you really want to spend 3 - 5 years of your life after a graduate degree to get a post graduate doctoral degree?
Second, you need to decide in what field.  
Third, in your country or cross border or overseas?  Online or traditional setting?
Fourth, start researching which schools meet your criteria.
NOTE: A big mistake some people make when doing research is not verifying that they have the requirements to even go to post-graduate school.   Or, some people assume they don’t have the requirements, and give up without doing the research.  Do not assume.  Do the research.  Answer the first four questions above, and then do research to see what’s out there.  Don’t cut off your potential using assumptions.  Research first, and then make your decisions.