To start the Forbidden Legend quest, head to Dragonsreach in Whiterun, and enter the room of the Court Wizard, Farengar. To the Southeast, there is another small room with a library. There are two bookshelves on the SE wall. There should be a book under a Flagon in the right bookshelf, second shelf from the bottom. The book is called "Lost Legends". Read it to start the "Forbidden Legend" quest.

Note: Forbidden Legend can also be started by reading a copy of the "Lost Legends" book in any other location, or investigating a cryptic message you found in one of the dungeons, such as when doing the "Under Saarthal" quest for the Mage Guild. Or it can also be started by entering Folgunthur and finding the body of Daynas Valen.

Open your quest journal:

Forbidden Legend
Long ago, the Archmage Gauldur was murdered, and his three sons were hunted down by King Harald's personal battlemage. The entire incident was covered up, their names struck from every record. But the legend survived. Perhaps someone still knows the truth of this ancient tale.

Investigate the Gauldur legend.

If you open the world map, you should see the quest marker on Folgunthur, between Solitude and Morthal. Fast travel to Solitude (if you have not been there already then take the carriage from Whiterun Stables), and head Southeast. Reach a small campsite with some tents. Read Daynas Valen's Journal. It will mention an Ivory Claw which he traced to a collector in Bravil, and an Amulet, which Daynas was after.

The quest journal should be updated:
The mage Daynas Valen spent his life searching for the truth of this tale, and came to Fogunthur with the key need to unlock its secret.
Objective: Learn the truth from Daynas Valen.

Head east of the campsite, and find the door to Folgunthur nearby. Enter Folgunthur. Head down, past the bodies of a Draugr and some adventurer. You will go past a keyhole requiring the Ivory Claw. It seems however, that the claw has already been used and the door opened, most likely by Daynas. In the next chamber, the pillar puzzle has also been solved already. Fight your way through some Draugr to reach what looks to be a big dining room, which will also have some Draugr enemies including a Restless Draugr.

Head NE, when you step on the grate it will open and you will fall into the water. Swim up and activate lever to close the grate and push you back out of the water. Climb the spiral stairs and cross the bridge. Find the body of Daynas Valen laying near a keyhole. Search the body to find the Ivory Dragon Claw, as well as Daynas Valen's notes. Your objective now is to read the notes, so locate them in your inventory under Books and read them.

Open your quest Journal:
Long ago, the Archmage Gauldur was murdered by his three sons, who stole his amulet of power and divided it among themselves. The brothers were hunted down in secret and sealed in tombs across Skyrim. To reclaim the amulet, I will need to seek out their final resting places.

Find the Gauldur Amulet Fragment in Geirmund's Hall
Find the Gauldur Amulet Fragment in Folgunthur
Find the Gauldur Amulet Fragment in Saarthal

Note: If you have already found any of the amulet fragments, the corresponding objective would be already marked as completed. For example, if you completed Under Saarthal for the College of Winterhold, you should already possess the Gauldur Amulet Fragment from there.

Folgunthur Amulet Fragment

Activate the keyhole near Daynas' body using the Ivory Claw to lower the bridge in front of you. Cross the bridge, battling through the couple of Draugr on the other side. There is a chest to the South East, but watch out for the traps. Head towards the corridor to the SW, and reach a locked gate with four levers near it, two levers on each side. This is a PUZZLE.

Lever Puzzle solution:

Locked Gate
L1         L3
L2         L4

Pull levers L3 and L2 to the UP position to open the gates. The other levers (L1 and L4) should stay in the DOWN position. Go through the now open Portcullises and down the stairs. Open the wooden door to reach a circular chamber. Two Draugr will rise from the thrones and attack you. Defeat them. Observe the grate in the center of the room, there are some stairs underneath it. There is a chain to the SW of the room, but when you activate the chain you set a dart trap. This is another PUZZLE.

Solution to puzzle:

On the North wall, to the right, there is a lever. Use it to open the door to a small chamber if it's closed. Observe the circular statue with three heads on it. Working counterclockwise, from the symbol closest to the door, the order of the symbols are Snake, Whale, Bird. Exit the chamber and go to the other door to the left of the northern wall. Use the lever to open the door. Working clockwise, set the pillar closest to the door to Snake, then the middle pillar to Whale, and the furthest pillar to Bird. Pull the chain in the main chamber and the grate will open. Head down the stairs, and past an area flooded with water. Beware of the Frostbite Spiders that are awaiting you. Head NW and reach a corridor where lots of Draugr will emerge out of their tombs and attack you. Once you defeat them, head to the rings door. This is a PUZZLE.

The door has three rings on it. Each ring can rotate to show one of three symbols: Wolf, Dragon and Hawk. The solution is found on the Ivory claw you got from Daynas' body. If you examine it carefully, you will see the order of the symbols. Set outer ring to hawk, middle ring to hawk, and inner ring to dragon. Activate the keyhole to open the door. Then go through another door into the Folgunthur Crypt. Enter a big burial room, and a boss battle awaits with Mikrul Gauldurson.

Defeat him,  then loot the Gauldur Amulet Fragment as well as the Writ of Sealing and a very good  sword, the Gauldur Blackblade, a Vampiric sword that absorbs health.

On the south wall, there are two keyholes Activate the one to the right, and exit the chamber. Head to the glyphs on the wall to learn the Word of Power Frost, Frost Breath. There is also a good piece of light armor and a valuable diamond ring in  a chest so make sure to loot them before heading out the door to the Southeast to exit to Folgunthur. Activate a sarcophagus lid on the NE wall and make your way back out. In the first room where you entered, the one with the solved pillar puzzle, there is a keyhole on the wall to the east which will reveal a chest with some treasure, so get it if you want, then exit Folgunthur.

Geirmund's Hall Amulet Fragment

Next, you want to go to Geirmund's Hall, which is to the east of Ivarstead. If you have not been to Ivarstead yet and cannot fast travel there, you will have to go to Whiterun and take the road east around the mountains. The way to Ivarstead is quite long. After you have reached Ivarstead, head east to Geirmunds Hall and enter it. Fight your way through a couple of Skeevers, drop down the hole into some water, open the Iron Door and defeat the Frostbite Spiders behind it. Head west to arrive at a flooded cave. Fight the Draugr there, including a restless Draugr. Notice the pillars. This is a PUZZLE

Solution to Geirmund's Hall Pillars Puzzle:
On the stairs leading to the flooded cave, observe the symbols to the left and right of it. There is an eagle and a whale to the left (when facing West), and a Fish and Snake to the right. Set the first pillar, closest to the stairs, on the left wall, to the eagle. Continue along the left wall and set the pillar near the Portcullis to a fish. Go back to the stairs, then head towards the portcullis again, this time near the right wall.  Turn the first pillar to be a fish (it should already be in that position) and the next one to be a snake. Then pull the lever to open the gate.

Go north into the cave and through a narrow corridor. Fight the Draugr on the other side, and climb up the stairs. On the altar observe that the corpse of Archmage Geirmund is holding a key. Take Lord Geirmund's key and open the iron door to the north. Beat the Restless Draugr and Frostbite Spider inside, and then take the stairs to the west. Kill a Draugr Wright and activate a lever to lower the bridge, then activate a second lever to lower the next bridge. Work your way through the corridor and past the Pendulum Traps, jump into the water and climb up the stairs leading to the coffin. Sigdis Gauldurson will emerge from the tomb.

He suddenly teleports away and summons illusions which look like him to fight you. Make sure to focus on Sigdis and not the illusions. The illusions can be identified by the blue glow. Also, Sigdis' helmet has horns on it while the dopplegangers' helmet does not. Once you defeat Sigdis, the illusions will be dispelled. Loot him to find the Gauldur Amulet Fragment as well as a writ of sealing and a Gauldur Blackbow which is a very nice bow that absorbs 15 points of magicka.

Exit through the corridor behind the tomb, to the east. Reach a room with a big chest that has some good treasure in it. Then continue along the corridor, activate a lever and a section of the wall will open, leading you back to the starting area. Exit the cave back to Skyrim.

Saarthal Amulet Fragment

The only way into Saarthal is by doing the first two College of Winterhold Mage Guild quests. If you go directly to Saarthal, you will arrive at a locked door that requires a key and cannot be picked. Instead, travel to Winterhold to join the Mage College. The following two guides will walk you through doing the quests there, once you are done with them you will have the third Gauldur Amulet Fragment in your possession:

College of Winterhold - Mage Guild Quest 1: First Lessons Walkthrough

College of Winterhold - Mage Guild Quest 2: Under Saarthal Walkthrough

Open your quest journal after getting all three pieces of the amulet:
I defeated the brothers and reclaimed the fragments of the amulet. Perhaps Gauldur's tomb holds the secret to restoring it to its original form.

Your objective now is to reforge the Gauldur Amulet. Open your map. Your destination is to the Southeast of Markarth. Fast Travel to the city or hire a carriage. Go SE and locate the waterfall, go under it to enter Reachwater Rock. Walk your way through the caves. An adventurer's body lays dead near a pedestal. Search the body, and grab the Ancient Edict and the Emerald Dragon Claw from the Pedestal.

Arrive at a Rings Door Puzzle. Examine the Emerald Claw. The combination, from outer to inner, is Bear, Whale, Snake. Several doors will open in front of you, go through them and arrive at yet another puzzle door. For this one, examine the Ivory Claw. The combination, starting at the outer ring and to the inner ring, is hawk, hawk, dragon.

Go down the tunnel, and reach a big chamber. Approach the Amulet Pedestal to the North, and activate all three of them, placing an amulet fragment on each one. Expect a tough battle now, against all three brothers. Once you have defeated them, take the Gauldur Amulet from the Altar.

"I defeated the undead brothers, located the fragments of the Shattered Amulet, and forged it anew in Gauldur's Tomb"

The Gauldur Amulet is a very powerful amulet, that increases your health, magicka and stamina by 30 pts. This is perhaps the most powerful amulet in the game, and a well deserved reward for completing this mission.

This completes the quest "Forbidden Legend"