There are currently 8 breathings in the game.
You can get each one by talking to a trainer and accepting him as your teacher.
Though, mostly what he will do is just fight with you to get new moves at specific ranks.
The overall primary breathing in the manga. It currently has 4 moves.
=== Beast ===yessir
One of the secondary breathings. It has 4 moves.
Another secondary breathing with 5 moves.
Wind also is a secondary breathing with 4 moves.
It's a secondary breathing. However, as of now it only has 4 moves.
Another secondary breathing. It obviously has 3 moves.
A secondary breathing that may interest you. It posses 4 moves.