The Biggoron's sword is VERY hard to get the first time. You have time limits on perishable items, and once you start, it's best to continue on until you've gotten the sword. You need the sun's song if you want to make it easier, and you must have Epona to make it work. Oh yeah, if you have blue fire, which you can find in the Ice palace, use it to unfreeze King Zora because you must talk to him. The Trading is much like Zelda: Link's Awakening. Give and item, get an item.

The Cuccos
First, what you must do is head to Kakariko, to the cucco lady. Talk to her to receive a pocket cucco. Use it to wake Talon, sleeping in the house near the stairs heading to Goron Mountain. Once you've done that, return the pocket cucco to the cucco lady and get a blue cucco, Cojiro, who is unable to crow.

The mushroom
If you take Cojiro to the Lost Woods, he will start to crow. Take the path left from the entrance, and you will find the owner of Cojiro. Wake up the guy and give Cojiro to him. In return, he will give you the odd mushroom.

The Odd Potion
The odd mushroom must be delivered to the woman quickly. If you take the path left from the boy on the stump, ride the magic bean to the bridge, jump, and head right to exit from Kokiri Forest quickly. Head to Kakariko on Epona. Enter the potion shop, and go through the doorway to the exit. Now, take the stairs up to the house (not the ladder) and enter to find the old woman. She will give you the odd potion.

The Poacher's Saw
If you return to where you met the boy in the Lost Woods, you will see a young Kokiri instead. She will give you the poacher's saw in exchange for the odd potion.

The Broken Goron's Sword
If you take the poacher's saw to the man standing in front of the tent in Gerudo Valley (West in Hyrule Field), he will give you a broken sword.

The Prescription
Now, take the broken sword all the way to the top of Death Mountain. Near the entrance to the great fairy, and the Fire Temple, you will see a HUGE Goron. He will be rubbing his eyes constantly, because of the recent eruption. Take the prescription that he hands to you. If you get him some eyedrops, he will make you a fine sword.

The Eyeball Frog
You must take the prescription to King Zora. He will not give you the eyedrops, rather, he will give you the main ingredient for the eyedrops, the eyeball frog.

The Eyedrops
You must be very quick now! Take the eyeball frog to the man in the laboratory in Lake Hylia. At first, he will think that the eyeball frog is for him. After he realizes that it is not, he will make you up a batch of eyedrops. They have no preservatives in them, so they won't last.

The Claims Check
Now, this is the toughest part. Quickly, take the eyedrops back to the top of death mountain and give it to the Goron. He will use it immediately and give you a claims check. He will also tell you to wait three days.

The Biggoron's Sword
Wait three days? Screw that! Use the Sun's Song six times to make three days go by. When that happens, show the Goron the claims check for the second claims check. Just kidding! He'll give you the Biggoron's Sword...finally! Now wasn't that worth it?