The easiest way to get promotions is to grind Treasure Battles. This will also net you lots of items and fight money as well.
I used Lucky Chloe and did her "coffee grinder to scoot" move over and over. If facing right, the move is
. After the first hit connects, you can keep hitting
and the move will loop.
Doing just this move, I went from Beginner to 9th dan/Fighter in a single run of 53 fights. Once you complete all the kyu ranks and get to 1st dan, it will start to take longer between promotion chance fights, just stick with it.