I have removed the existing flooring down to the plywood. The plywood isn't very stiff and I am concerned the tile will crack. I am installing natural stone tile (Travertine) and would like to know what subflooring material to install between the plywood and the tile. I have heard cement backer board (I thought this only went in bathroom walls), OSB, plywood, particle board, etc. I have also heard I need a bond breaker (orange waffle looking stuff - can't recall the name). Any suggestions?

You need the floor thickness recommended by the manufacturer of the tile you are using.
Concrete backer board, if placed properly over the joists will definitely add strength and give a good surface for the mortar/adhesive. But depending on the thickness of the existing floor you could need an additional layer of plywood or whatever is acceptable.
I'm not familiar with the bond breaker requirement but I am sure someone here is.

You need at least a 1/4" backer like Hardibacker for this installation. They also make a 1/2" backer if your sub is not already 5/8". The backer gets glued and screwed with special screws. Use modified thinset to glue the backer to the floor and then butter all joints after laying down fiber glass tape on every seam. Wonderboard would perform similarly but is a bit messier to handle.