After you reach Matango, you learn from King Truffle in Fung Castle that there is a white dragon in a nearby cave.
Level Up Magic
Sylphid's magic is helpful here, so have Popoi level up Sylphid's spells.
Find the White Dragon
King Truffle opens the path to the White Dragon's cave. Go to the room of Fung Castle where Watts is talking to a mushroom guy, and use the southeast exit. Then follow the path outside to the next area.
There are various enemies in the area such as Water Thugs and Steamed Crabs. Just keep going north until you reach the cave.
In the cave, go up the stairs and through the door. In the next area, you can use Thunderbolt on the Pebblers to get rid of them quickly. Otherwise they take a while to defeat with weapons.
Go around the path until you reach some stairs to the upper area. Go up those stairs and go south along the ledge to reach a stalagmite sticking up out of the ground. Stand above it and use the whip to go to the left. Go down the stairs and go to the left, and down the stairs to a lower level.
After dealing with the Kimono Bird and any Pebblers it creates, use the whip to go to the right across the gap. Go down the stairs just past the gap. Continue along the path and go to the left when the path splits. Go up the stairs and go to the left, then go down the stairs at the end of the path. In the next area, go to the right, and go up the stairs.
In the next room, go to the left to find a crystal orb. If you use Sylphid's Analyzer spell on it, it says to use Gnome's magic. You have to use Earth Slide. Go back downstairs, then go south. A path has opened down there that was blocked off by stalagmites before. Go down the stairs after going all the way south.
Get past the Kimono Birds and go into the cave entrance to the left.
In the inner cave area, Great Viper attacks. The most important part of the strategy is to use Thunderbolt on it as often as possible. It moves quickly, though, so it's difficult to target or hit it with weapons. It can bite your characters, pygmizing them. Use your Midge Mallet (which you hopefully got from the Elder in the Dwarf Village) to go back to normal if you're pygmized. The Great Viper can even swallow pygmized characters.
When you defeat Great Viper, you get the Sword's Orb. Afterward, go north into the cave entrance there. There are no enemies inside. Go through the cave and talk to the white dragon. It's still a baby. King Truffle will take care of the dragon. He names it Flammie. He asks your opinion of the name, but before you can answer, he decides that you'd probably agree no matter what, so he settles on the name Flammie.
Next he tells you to go to the forest to the south and use Cannon Travel there to go to Kakkara Desert. Be sure to buy more Faerie Walnuts and things before you go. Then go south from Fung Castle, east to the cave entrance, then go through the cave to reach Upper Land Forest again. Go southeast to reach the Cannon Travel guy. Pay to go to Kakkara Desert.