Who taught Tsunade her medical jutsu? The greatest medical ninja should have learned it somewhere. 


  Tsunade's skills extend beyond the typical medical ninjutsu. She earned fame during the Second Shinobi World War for being the only person able to make antidotes for the poisons of Sunagakure's Chiyo. Similarly, she can make sleeping potions that are undetectable to even the most skilled ninja. She is knowledgeable enough to perform autopsies and examine damage on a cellular level.


Source from Narutopedia.
There's no background information explaining how Tsunade became a medical-nin or how she learned medical ninjutsu. She didn't have a medical specialist mentor. Maybe she did have one, but there's no source or background information to prove it. 

There is a possibility that she learned medical ninjutsu all by herself, as she often read other clans' medical books to solve particular problems.