Two spells that are practically necessary for this trip are Float and Wall. When you enter, you will notice some little red patches on the ground. If you step on them, HP goes down. So use Float on the party.
Head down and right to get a HiPotion from the chest, and go north to get a Life and through the wall on the left to get a Ether1. Return to the main part and go south then left. Go through the door. You�ll have to use Float again since it wears off after each floor. Continue along the path and take the bottom path to get a HiPotion. Return and go up. Take the path on the left for a Cabin. Go right and down the stairs. Go left, down, and through a secret wall on the left. In the first indent, there is a HiPotion, a Venom Axe and a Kikuichi. Continue further along the path to the second indent, where you�ll find a Bestiary. Continue to the last indent, and go north to open a chest, guarded by 5 Warriors, which contains the Defense Sword.
Return back and left until you see a podium. Ignore it for the moment, and head south and left to get a Life. Return and step on the podium. Get the Ether1 to your left and the 6000 Gil down the stairs. Continue right to get 5000 Gil, and further to get a Bestiary. Step on the podium to your right. Step on it again and go down to get the Rat Tail. Go back and through the podim. Stock up on items, weapons etc. In the top left house is a Bestiary and a Life. Save and head to the Library beneath the room you are in. Head left into the basement and step on the next podium. Talk to the Queen to initiate a battle.
Asura:
Remember I said Wall was necessary? Well now is the time. Cast Wall on Asura so that she reflects all of her healing spells on to you. After that, just pound on her. Keep healed though.
Once you win, you will gain Asura as a summon. If necessary, return to the save point and heal. When ready, return to the room. Talk to the man to fight him.
Leviatan:
He starts the battle with Big Wave, so immediately heal. Have edge Throw, use Blitz or attack, have Rydia use her highest Bolt spell, Berserk Cecil and Rosa heal.
When you beat him, Leviatan is yours. Now you�ve done that, leave The Land of Summoned Monsters.