Monomi game is basically like this.
Press Square to draw magic lines.
Complete a circle, and it will set off a magic spell, so your aim is to trap monobeasts inside the circle.
If you draw a big enough circle (and managed to link the circle together), you can cast off a bigger version of the spell.
For armored Monobeasts, the ones that will take a while to bring down, you can press "O" to jump on them, followed by square to do a somewhat mid-air attack, which will strip their armor down and make them easier to take care of.
For every magic circle that you complete, or when you get damaged, you gain some meter towards your "Usami bar". Fill it to 100%, and you will be able to transform Monomi to Usami, which gain stronger magic and at the same time invulnerable against enemy attacks, which makes Usami ideal to use during boss battles.
The better equipment you equip, the stronger your character get, the more your Usami gauge will last as well.
If you want to hunt equipment (there are 60 in total), having the grass skirts help a lot as they boost the chance of present boxes popping up.