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.
(if you can't see the notation, the move is down-back rightkick rightkick)
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 you will start at 1st dan and go up.