Battle performance
Your performance during a match is set by your champ's base power (which depends on its level and equipment), training results, and using an elemental weakness. Each of those numbers will total your battle performance. If your battle performance is higher than your opponent's, you will win.
Elements
There are three different elements. Earth is stronger than Fire, Fire is stronger than Shadow, and Shadow is stronger than Earth. Using an element that your opponent is weak against results in ten to fifteen additional power in your overall attack. Because there is no definite way to tell what element your opponent is, have a champ that is strong in each element.
Training combos
During training, a combo is made by one straight slash through multiple balls. Try creating straight clean slashes to get combo bonuses. You can also get a screen-clear bonus when you slash all the balls in a single wave. Make sure to avoid the bombs. When slashing at the balls, keep your finger on the screen and drag a path through them.
Various Tips and Tricks
Elemental exploits do help a bit. There are three different elements in this game: earth, fire, and shadow. Earth beats fire, fire beats shadow, and shadow beats earth.
During fights elemental weakness adds around 10-15 power to your overall attack power.
The outcome of a match is determined solely through three factors:
Your champ's base power - affected by how high your champ's level is and whatever equipment they have on.
The training results - rain regularly to keep your character ahead of the competition. At least once between each fight.
Hitting an elemental weakness.
Straight slashes nets you higher combos during training.
Avoid the bombs while swiping at the balls.
Battle performance...