After defeating Hydra and getting the Sword, you can go to Laconia and get the Eye of the Graeae.
Defeat the Cyclops
With the Sword, you can defeat the Cyclops that is blocking the path to Laconia. To reach the Cyclops from the Nymph's room in Peloponnesus where you got the Sword, leave the Nymph's room and go to the right until you reach a dead end. Go through the door and follow the path until you are in the Forest of Thorns.
In the Forest of Thorns, go to the right until you reach a dead end, and go through the door there. Follow the path until you are in the main forest.
Go to the left and go into the third door (it's the second to last door). You will see an old woman in a small forest. Go to the left to reach the next area. There are three tree branches to the left. Jump onto the third one (just before the rocks) and hit the tree trunk with your sword. An Ambrosia will fall from above. Take it to increase your maximum health.
Go to the left and avoid the bat (you can throw a flame from the Staff of Fennel to kill it while it is still in the tree), then continue going to the left and the Cyclops will appear. Stay on the tree branches to avoid the rocks that the Cyclops throws. It can be difficult to defeat the Cyclops. If you are in the tree branches, wait for the Cyclops to throw a rock, then go down and hit the Cyclops's eye with the sword (the Staff and Club won't harm it), then quickly run away. If you are on the ground near the Cyclops, walk away from it to avoid the rock that it throws onto the ground. If you are on the ground far from the Cyclops, walk toward it to go under the rock that it throws past you, then jump over the rock after it bounces, then quickly jump up and hit the cyclops's eye with your Sword. As soon as you hit it, run away so that it will throw the rock on the ground and won't hit you with it. Keep going back and forth this way. When you hit the Cyclops's eye four times, it dies.
After the Cyclops dies, go to the left and enter the door to reach Laconia.
Get the Trojan Pitcher
In Laconia, there are Scorpions on the ground, as well as birds that will drop rocks on you. There are also some monkeys that will jump toward you if you get close.
Go to the right until you pass by a small Sun monument. There is a broken pillar to the right of it, then an unbroken pillar to the right of that. Kneel at the base of that unbroken pillar and hit it with your sword. The Trojan Pitcher will fall from above. Now, whenever you use a fountain, you will also fill the Trojan Pitcher with nectar. If you equip the pitcher and press SELECT, it will refill your health to full, but after that, you will have to use a fountain again to refill the pitcher.
Get the Ocarina
You will need 60 Olives to get the Ocarina, so collect them now if you don't have enough. You can go down the stairs from the upper part of Laconia to find Satyrs. They are relatively easy to fight, especially if you use the Staff of Fennel to throw flames at them.
From the Sun Monument near where you got the pitcher, go to the left of it, and hit the base of the first unbroken pillar. A pit will open in the ground. Go down, then go into the door, follow the path, then shoot a flame from the Staff of Fennel to light up the dark cave. Go to the right and go into the door to reach Poseidon's temple. Talk to him and pay 60 Olives to get the Ocarina.
Leave Poseidon's temple and use the Staff of Fennel to light the dark cave, then go to the left and go through the door. There will be a girl in the hallway whom you can talk to. Go left from the girl, and you will come out of a doorway in the lower part of Laconia.
Go to the left and go down the stairs. Then go all the way to the right until you reach a dead end. Go into the bottom-right door. Follow the path, then use the Staff of Fennel to light the dark cave.
Go all the way to the right and you will encounter another Cyclops. Equip the Sword, then go toward the Cyclops, then run away when it throws a rock, then run toward it so that you run under the next rock that it throws. Carefully jump and hit its eye, and run away so that it throws its next rock on the ground. Keep going back and forth this way, hitting Cyclops's eye when you can. This Cyclops will die after you hit its eye six times with the Sword.
Take the Ambrosia that appears after you defeat the Cyclops, then go to the right and go into the door to find a fountain. Kneel at it to refill your health and fill up the pitcher.
Defeat the Siren
Leave the fountain room and use the Staff of Fennel to light the dark cave. The Cyclops will not be there. Go all the way to the left and go through the door, then follow the path to return to Laconia.
Go to the left until you reach the next area. Go up the stairs to reach the upper part of Laconia, then go all the way to the right until you reach the ocean. If you don't have the Ocarina equipped, press START and choose the Ocarina. You can press SELECT to use it. While standing at the ocean in the far-right part of Laconia, use the Ocarina to summon a dolphin. Hop onto the dolphin and go to the right. Flying fish will jump out of the water. You can avoid them by getting close, then swimming away, because this will make them jump and then disappear into the water. Or you can just hit them with your sword to kill them.
Keep going to the right until you reach an island. Jump off and go to the right, and you will find the Siren. If you don't already have the Harp equipped, press START and choose the Harp. You can press SELECT to use it.
Use the Harp whenever you are unable to move. Then hit the Siren's face three times with the Sword to kill her. After she dies, go through the door that appears.
Defeat the Graeae
Use the Staff of Fennel to light the dark cave. Then go to the right, and the three Graeae will attack, shooting magic beams at you. If you kneel on the raised step to the left of the Graeae, their magic beams will bounce off of your shield. Also, if you hit a Graeae before it can shoot a beam, it will disappear without shooting a beam. This will allow you to only have to dodge two beams instead of three. After the last one dies, it drops the Eye of the Graeae. Take it.
Now that you have the Eye of the Graeae, you can reveal hidden doors in certain places. One of these hidden doors is in Argolis.