You can get the Biggoron Sword as soon as you're an adult, although, it's better that you get familiar with the territory around you first.

Whatever you do, don't waste your 200 Rupees on the Biggoron Sword that can be bought in Goron City. It will break after eight hits. Instead, if you want the badass sword, follow the directions below.

As adult Link, go to Kakariko Village and speak to the girl whose chickens you rescued. She will give you an egg which will turn into a POCKET CUCCO. Use the POCKET CUCCO to wake up Ingo sleeping in the house in the village. Play the Sun's Song to have the sun rise again and take the CUCCO back to the girl. She will give you COJIRO.

Now head to the Lost Woods from Kokiri Forest. Go into the door to the left. Present the sleeping man with COJIRO. He will, in turn, give you the ODD MUSHROOM. He gives you three minutes to get the mushroom to the old hag in the potion shop. Back to Kakariko Village.

Exit the Lost Woods and Kokiri Forest. Ride Epona like the wind to Kakariko Village. Enter the Potion Shop, then take the back entrance out. Climb down the ladder and move up the stairs. Enter this Potion Shop, speak with the hag, then present her with the ODD MUSHROOM. The hag will give you an ODD POTION.

Head back to the Lost Woods, right where we met the guy before. He's gone, but give the girl the ODD POTION. She will give you the POACHER'S SAW.

Head out to Hyrule Field and get on Epona. Ride her to Gerudo Valley, the westernmost point of Hyrule Field. Use Epona to jump the broken bridge and speak with Master Craftsman. Present him with the POACHER'S SAW. He will present you with the BROKEN GORON SWORD. You need to take this to Biggoron on top of Death Mountain.

Here, I suggest playing the Bolero of Fire to get to the Fire Temple, then exiting out the top entrance to Death Crater instead of the one that takes you to Goron City.

Up here, present the Biggorn with your BROKEN GORON SWORD. He will want to repair it, but his eyes are all messed up because of the smoke. He needs eyedrops from King Zora, and he will give you a PRESCRIPTION for them.

Head to Zora's Domain and present the King with the PRESCRIPTION. He will give you an EYEBALL FROG to deliver to the man in the lab at Lake Hylia. They will spoil, so be warned.

Exit out of Zora's Domain and into Hyrule Field. Call Epona and ride like the wind to the entrance to Lake Hylia. Have Epona jump the two fences and ride straight up to the lab. Go inside and give the man the EYEBALL FROG. He will give you the WORLD'S FINEST EYE DROPS, and he'll give you four minutes to deliver them. RUN!

Hop back on Epona, exit to Hyrule Field. Get to Kakariko Village and take the long way to the very top of the mountain where you will present these DROPS to the Biggoron.

You'll get a CLAIM CHECK for the Biggoron's Sword. Play the Sun Song 6 times (you need 3 days to pass). You now have the BIGGORON'S SWORD!