This first bonus round will start as soon as you defeat Katana in the Riverside level. All you have to do is destroy the barrels which roll on screen by attacking them, the more you destroy, the bigger the bonus is as the end. Destroying them will get you experience points at the end of the bonus game.
The second bonus round will start as soon after you beat the Factory level. This game is exactly the same as the last one, with you destroying the barrels as they appear on screen. You can get a certain number of experience points by destroying numbers of barrels.