Once the conversation ends head forward and perform a jump with a swing.
This way you'll land on the platform below. There you will find two experience chests as well as one with a Gorgon Eye [!!!].
Climb the vertical wall and you'll reach spot at which some basic enemies will attack you. Continue along the path to reach a save point.
Once there, you can first turn left and collect the experience chests (four) an afterwards go back and go right, jumping along the ledges (however it's not necessary, as you'll have to pass by the chests mentioned before anyway).
Perform a couple jumps and you'll end up by the locked door leading to the Palace of Hades. You will be attacked by basic enemies there, however this time in a bigger group.
Once you're done with the last one, approach the spot to the fight of the locked passage. There you will find a rope - use it to get to the other side (during the crossing you'll be attacked by enemies approaching from both sides).
You will find a chest and a mysterious note lying on the ground. Waste no time and head to the cave before you. After jumping down, go backwards a little [RS], this way you'll find two experience chests.
A little further you will come across a gate. Open it by quickly tapping L1 and R1 and get ready for a fight. During it you'll be attacked by Gorgons, monsters which can turn you into stone. During the fight focus on them (there will be two, one after another, never at the same time). They are more demanding than the normal enemies, however their attacks can be blocked (including the stone-turning attack, which also requires pressing one of the basic buttons - failure to do so results in petrifaction; you can free yourself from it by quickly moving the left analog stick). In order to finish off the Gorgons, you have to first press two basic buttons and afterwards quickly tap Circle (this way killing the enemy will also petrify the other enemies, which will allow you to eliminate them quickly).
A Gorgon, apart from the previously mentioned moves, can also grab and squeeze you with its long tail. In order to free yourself, you'll have each time to quickly tap L1 and R1.
After the fight, go out through the next door (once again, quickly press L1 and R1) and go forward, to the next cave.
By the entrance, there are two chests with health and magic - use them if you need to. Go to the big room (you'll pass a blocked passage on your way) in which you will have the chance of obtaining Apollo's Bow.
In order to get it, jump onto the platform [LS] and from it onto the ledge [RS]. You will find a pot filled with Bramble - throw it down.
Afterwards push it in the direction of the Cerberus cage. The beast will shoot it with a flame and incinerate it, allowing you to fly up and open the cage. But before you do that, you have to return to the platform and lower it once more, by jumping onto it. Now quickly run through the room, jump onto the flame with your wings spread out and fly up. Pull the lever and jump down to defeat the new enemy.
The Cerberus isn't the hardest enemy out there. His attacks include a triple fireball, a firebreath (close distance) and traditional biting. Attack him and initiate a QTE at the end (two basic buttons) to jump onto his back.
During the "ride", you will have the ability to launch two types of attacks: a strong, short range with Triangle and a more long range one with Square. Rush onto the enemies until you reach the Bow. Burn its owner and get rid of the Cerberus by quickly pressing Circle.
Collect your new weapon and get rid of the enemies (some of them will be on a higher platform, so use the Bow). Your new weapon also brings the ability to unlock previously blocked passages (those covered with withered vines) and incinerate enemies, even a couple at a time (remember this, as it makes fights easier.
Before you go back to the blocked passage in the cave, notice that on the floor from which you've pulled the box down is another door. Unlock it by burning the vines and jump, using the fire to glide. You will find a chest with a Phoenix Feather [!!!] inside and an experience chest as well.
Now you can return to the blocked passage and go through it (save your game a little further). Get out of the cave and you'll arrive at an open area with more vines. Burn them to unveil a chest with a Minotaur Horn inside [!!!]. Nearby you will also find chests with mana and health.
The road leads right. Follow it and you will reach a room where Harpies will attack you. At this point in the game, with your main weapon not upgraded, it's best to just use Circle to get rid of them. Grabbing and tearing them apart is a quick and safe way to kill them (of course you can also grab them while in the air).
Their main attack is "bombarding" from above. Dodge the attack using the right analog stick. It can also turn out that one of the monsters will grab you. In such case, quickly move the left stick to free yourself.
Once you upgrade your weapon, you can attack them by jumping into the air and using Square with the L1 button pressed. This way you will perform a strong attack which will hit many enemies at the same time.
Go right, in the direction of the gap. Spread your wings there and glide to the ledge. From there jump to the narrow ledge and from there jump onto and grab a vine wall. Climb up.
Once you reach the top, head left and jump through the waterfall. This way you will reach a spot at which you'll find a chest with a Gorgon Eye [!!!] inside. Now head right.
After reaching the end, climb up and you'll finally be able to stand on your legs again. Not for long - do a double jump to grab the ceiling. Go along it to the end, jump down onto a ledge and glide towards another vine wall. Climb up to reach your destination.