You can get Biggoron's Sword as soon as you pull the Master Sword from the pedestal the first time, but only if you have Epona (read how to get Epona), and you planted a Magic Bean in the entrance of Dodongo's Cavern. It helps to have another bean stalk near the Lost Woods entrance, but it's not completely necessary.
Before attempting this quest, you should also get the hookshot from Kakariko Village. Check the Temple of Time section to find out how to get the hookshot.
Unfortunately, if you haven't learned Epona's song and planted a magic bean at Death Mountain Trail before becoming Adult Link, you will have to wait until you can go back to the past, which you can only do after you have completed the first Temple. Biggoron's Sword is not necessary to beat the game, but it makes things easier.
Once you have Epona and have planted a Magic Bean in the entrance to Dodongo's Cavern, you can start the quest.
First stop: Kakariko Village
Talk to the Cucco girl and she'll give you a Pocket Egg. Turn it from night to day (or day to night to day) to hatch the egg, then go into the house where Talon is sleeping (it's the non-brick house near the well) and use the Pocket Cucco on Talon to wake him up. Give the Pocket Cucco to the Cucco girl and she gives you Cojiro.
Second stop: Lost Woods
Ride Epona to the Kokiri Forest and enter the Lost Woods. Make a left from the entrance. That creepy-looking guy is sitting by the tall stump. Show him Cojiro and he gives you the Odd Mushroom. This item will go bad after a few minutes, so you have to hurry to the next stop. If you planted a Magic Bean in the area that overlooks the bridge to Kokiri Forest, go to it and ride the bean stalk, then jump onto the bridge and exit the forest. If you didn't plant the bean, go back to the first area of the Lost woods and take one of the wrong exits. Go out to Hyrule Field.
Third stop: Kakariko Village
Ride Epona to Kakariko. Turn it to daytime if it's night. Run to the shop, which is near the entrance to the Death Mountain Trail. It will be closed at night time, so make sure that it's day time before you start running toward it. Inside the shop, there is an exit at the left side of the counter. Go through it. Jump off of the ledge outside and run up the stairs. Go into the shop here and give the Odd Mushroom to the woman. Make sure that you are close to her and facing her, or she won't see that you are offering it. In return, she gives you the Odd Potion, which doesn't spoil.
Fourth stop: Lost Woods
Ride Epona to the Lost Woods, and go to where the creepy guy was. There's a Kokiri girl in his place. Apparently he didn't last long enough for you to give him the potion.... Anyway, give the girl the Odd Potion and she gives you the Poacher's Saw. Go out to Hyrule Field.
Fifth stop: Gerudo Valley
You have to ride Epona for this step. Ride to Gerudo Valley. The entrance is in the west of Hyrule Field. Slowly trot over the wooden bridge, and then use carrots and gallop over the broken second bridge. If you do it correctly, Epona will jump over the gap. Dismount and give the Poacher's Saw to the carpenter. He gives you the Broken Goron's Sword.
Sixth stop: Death Mountain Trail
Ride Epona to Kakariko, and go to Death Mountain Trail. When you reach the entrance to Dodongo's Cavern, step on the bean stalk and jump off of it when it starts to go back the way it came, which will happen near the beginning of the trail leading to where the eruptions start. Go to the base of the wall that you can climb (you can avoid any falling rocks by watching out for shadows). Be aware that there's a Tektite above. If you wait just beyond the base of this wall, the Tektite will fall. You'll want to lure it to you this way and kill it so that it doesn't hit you while you're trying to climb. Kill the skulltulas with the Hookshot and climb up. Walk to the area that is to the right of the entrance to Death Mountain Crater, and Biggoron will stand up. Show him the Broken Goron's Sword and he'll give you the Prescription.
Seventh stop: Zora's Domain
Ride Epona to the entrance to Zora's River. Before going in from Hyrule Field, turn Epona so that she's facing away from the Zora's River entrance. Now make your way to Zora's Fountain, which is where you found Lord Jabu-Jabu in the past. You'll notice on the way that the whole place is frozen over, and King Zora is trapped in a block of red ice! We'll have to fix that.
To rescue King Zora, you will have to go to the Ice Cavern to catch some blue fire in a bottle. You don't have to complete the whole cavern, though. You can quickly go in, get the blue fire, and go out, if you choose. Just go from the entrance straight back to the back room, get the blue fire in a bottle, and go back out. Refer to the Ice Cavern section for more information.
Use the blue fire next to King Zora to melt the red ice. Then get on the platform and talk to him. In thanks he'll give you the blue tunic. If you got the Iron Boots in the Ice Cavern, you can go back to Zora's Fountain now, put the Iron Boots and blue tunic on, and sink to the bottom of the fountain for a piece of heart. Or you can do that later.
Then, if you're ready for a countdown, go back to Zora's Domain and show King Zora the prescription. He gives you the Eyeball Frog, which will spoil in a few minutes, so hurry.
Eighth stop: Lakeside Laboratory
To get out of Zora's Domain quickly, make a left from King Zora and jump off the right edge of the waterfall at an angle. You get hurt, but you save a lot of time. Walk across the ice and climb up the ladder that goes up to the exit of Zora's Domain, and exit.
Jump into Zora's River and run through the water out to Hyrule Field. Epona should be still waiting where you left her. Climb into the saddle and head for Lake Hylia. Use all but one carrot, then when a second carrot appears, use it. Repeat this to stay at top speed. You'll want to head around the left side of Lon Lon Ranch to get there. You'll see a fence in your way in the distance. There's a gap in it to the right. When you're near the Lake Hylia entrance there's another fence. Stay a bit to the left to avoid it. To make it over the fences to Lake Hylia you have to use carrots, so make sure you don't run out of them before you get to the fences.
In Lake Hylia, make your way in carefully, because there's a little slope on the way in that might make Epona stop if you hit it dead on, so stay near the wall. Once you're at the door of the Lakeside Laboratory, dismount and run inside.
Show the creepy guy the Eyeball Frog, and he'll give you the Eyedrops, which is another item that will spoil in a few minutes.
Ninth Stop: Death Mountain Trail
Quickly run out and climb into the saddle. Use carrots and make your way out of the lake, making sure to stay near the wall to avoid the slope that might be too steep for Epona.
Once out of the lake, keep using carrots and you'll get over the fences. I like to stay to the right of Lon Lon Ranch on the way to Kakariko. Be careful going over the bridge leading to Kakariko: if you go over it quickly with Epona at a bad angle, she might stop. At Kakariko, dismount and go in.
Run to the entrance of Death Mountain Trail.
Use the bean stalk just outside Dodongo's Cavern as before to quickly get up to the path with the eruptions. Remember to run out of the way if you see shadows on the ground. At the wall that you can climb, remember to stay a little distance from the wall to lure the tektite to land near you so you can kill it. Then use the Hookshot to kill the skulltulas on the way up. Quickly walk up to Biggoron (he's to the right of the entrance to Death Mountain Crater) and give him the Eyedrops. He gives you the Claim Check.
Turn it from day to night (or night to day) three times in a row. You'll have to wait for night to fall or day to break before playing Sun's Song again, because otherwise the song won't do anything. Show Biggoron your Claim Check and he'll give you Biggoron's Sword. (If it's not ready, just keep playing Sun's Song and checking with him until he gives it to you).
Now you have Biggoron's Sword! It's such a big blade you have to wield it with both hands, which means that you can't use your shield while holding it. You can still use the shield if you aren't holding the sword in your hands, though, so if you're holding the sword, stay still until the A button says "Put Away" to put the blade away, and then you can use the shield.
If you hit R while holding the sword, you'll hold the blade in front of you, but it doesn't block anything, unfortunately. This blade is twice as powerful as the Master Sword. It may have taken a while to get it, but it's worth it!