What does MS in Pure or applied Math look in top PhD statistics application
I went from an MS in pure math to a PhD in statistics. My transition was just fine. Many doctoral programs in statistics, especially those at "elite" universities, tend to actually be rather math heavy. You will be well suited to pursue questions in mathematical statistics and probability theory. A robust understanding of linear algebra and real analysis are always helpful in statistics. 

You will need to make sure that you have a background in at least a few things:


Be sure to perhaps take an introductory stats course if you can. This will allow you to learn the basics of the terminology.
Become competent in some programming languages. C++ is good. Python is good. R is great. You may struggle in some classes if you are completely inept at programming. 
Try looking at the backgrounds of some of the students at your desired schools. LinkedIn or the school website can sometimes give you this information without excessive work.
As for research experience, every bit helps. It would maybe not be a disqualifier if you have no research, but you should look into how feasible it would be to join in on a project. This being said, the biggest thing a committee would look for would be potential to do quality research. This can be shown in more ways than just straight up research. 


Also, may I second @astronat's thoughts on going directly to a PhD if you are so inclined? It's not a solution for everyone, but it can speed up the process of obtaining the terminal degree sometimes. Note that going directly for a PhD is more likely to allow you to obtain funding immediately for graduate school.