I found it.
At the top, just below the Logout button, there's a link "RPG". If you click on it, a dropdown menu will appear. At the bottom of the dropdown menu there's a link to the "Monster Arena", where you can battle monsters
EDIT#1: Another question.. what does depositing into the bank do?
EDIT#2: It seems excessively expensive to heal yourself at the training center. 54 Gil just for healing 24% of your HP! Sheesh. I hope you automatically regenerate HP over time, cause I'm not about to waste that much money for that little just yet
EDIT#3: The healing cost goes up as I get less and less HP it seems. Oh and, why don't you get Gil for killing monsters? In every FF game you get Gil for fighting, but not here :/ hehe
EDIT#4: I bought a potion for my hard-earned 100 Gil, and went into battle with a monster to use it (it gives 150 HP in battle, only 100 outside). It gave me 0 HP and I died. WTF? I hate this RPG game