After completing the first College of Winterhold quest, "First Lessons", you unlock the next quest in the series, "Under Saarthal". Open your quest journal:

Under Saarthal
Tolfdir has asked the apprentices to meet him at Saarthal, the site of an ongoing excavation by the College

Meet Tolfdir outside Saarthal

Feel free to explore the College of Winterhold, or, if you are eager to start the quest, head to the starting location. 

The Saarthal excavation site is Southwest of Winterhold. You can fast travel from the college to the city to save some time. While in the city, make sure to visit Birna's oddments to unload excess inventory and make room for the treasures you will find in the Saarthal ruins. Then, from the center of the city, cut Southwest between the two mountains. On the way you will meet Tsalgar the Wanderer, a bard who will sing a song for you upon request.

Reach the excavation site and talk to Tolfdir. Before going in, you can browse what he has to sell, he carries some spellbooks and robes. He can also train you in Alteration magic. After you are done, tell him you are ready to go. He will tell you to stay close to him while you are inside. Follow Tolfdir through the door and down  the ramp.

Ask Tolfdir what he wants you to do. He will ask you to assist one of his scholars, Ariel Gane, with his work cataloging and uncovering the ruins. Find Arniel Gane. Follow the tunnel with the lanterns, then down a ramp, and go through a bridge and another tunnel, to finally find Ariel Gane in a small room. Talk to him. Tell him Tolfdir sent you to help him. He tells you he has only looked in a portion of this section and asks you to explore the chamber north of here. He wants you to round up the rest of the magical artifacts so that he can catalog them.

Your quest journal will now be updated:
I've been asked to assist Arniel Gane in his work at the excavation in Saarthal by collecting enchanted objects scattered around the ruins

Search for magical artifacts (0/4)

Head north in search of the artifacts. Locate the artifacts throughout the corridors. There should be arrows pointing to them. Pick them up. You have to find 3 enchanted rings and a Saarthal Amulet. The three rings will be on the ground. The Ancient Amulet will be on some sort of door. Activating the amulet triggers some sort of trap! All the exits shut down. Tolfdir appears to check up on you. Tell him you are trapped. He will tell you to use the Saarthal Amulet to escape the trap. The quest log is updated again:

"While helping in the excavation in Saarthal by  gathering enchanted items, I somehow triggered some sort of trap. It has revealed an unexplored section of the ruins, and Tolfdir would like me to follow him further in."

Bring up  your inventory and equip the amulet. This will cause the section of the wall from which the amulet was taken to resonate. Fire a spell at the wall. This will blast open the section of the wall. Follow Tolfdir.through the opening

When you reach a chamber, suddenly the ghost of Nerien appears. He will tell you that you have set up a chain of events that cannot be stopped. Judgment will be passed based on your actions to come and how you deal with the dangers ahead of you. This warning is passed because the Psijic order believes in you. You alone have the potential to prevent disaster.

The ghost vanishes. Talk to Tolfdir about the apparition. He did not see it. Tell him it spoke to you, about some danger ahead and the Psijic order. He will tell you that's odd, since the Psijics have no connection to these ruins and noone has seen them in ages. He suggests looking for answers inside these coffins.

Your quest panel is updated:
"While in an unexplored area of Saarthal, I received a vision from someone in the Psijic Order saying that there's something dangerous in Saarthal. Despite this, Tolfdir wishes to forge ahead and has asked that I follow him"

Follow Tolfdir. Suddenly some Draugr appear! Defeat them, then keep following him. Activatet the lever downstairs to lift the gate. You will reach a big room. More Draugr will attack. Help Tolfdir kill them. After you kill the skeletons,  your objective is to find the danger within Saarthal. Activate the two chains on both sides of the door, and enter Saarthal. Be sure to save the game before doing so as many dangers lay ahead!

From the entrance, head east, and then south. Climb up some stairs and open an iron door to the east. Keep heading east, then turn north. Beware of traps and skeletons.

You will reach a section with six pillars. This is a PUZZLE

Each pillar has 3 possible symbols: Bird, Snake, and Fish/Whale

solution to Saarthal 6 pillar puzzle:
Look closely behind and above each pillar. The correct symbol will be there!

Facing the lever and the gate, in the room that's most  back, set the left pillar to bird and the right pillar to whale.

In the middle room, still facing the same direction, set the left pillar to snake and the right pillar to bird.

In the lever room, again facing the lever and gate, set the left pillar to whale and the right pillar to bird. 

The pillars should look like this:


Pull the lever. If the pillars are in the incorrect position, you will set a trap. If they are in the correct position, the gate will open.

Go through and reach another iron door. Enter the chamber. Be prepared for a tough fight with a Draugr Wight.

Climb up the stairs and go through yet another iron door. You will arrive in a room with 4 pillars. This is another PILLAR PUZZLE

Four Pillars Puzzle solution:
At first glance, the puzzle seems very easy! Two sets of symbols, one in the back, and also above the pillars themselves, both pointing out the correct position of the pillars. But when you turn one of the  pillars, others also turn undoing the position you have previously set them to!

here is the answer. Stand behind the lever, facing the lever and door (look at the compass, you will be facing SOUTH)

We will number the levers like that:

Pillar-4 Pillar-2
Pillar-3 Pillar-1

The number near the pillar denotes how many other pillars it rotates when you turn it.

Start by turning Pillar-4 to WHALE.
Pillars 1,2, and 3 will also rotate.

Next, move pillar 3 to SNAKE
Pillars 1 and 2 will also rotate

next, rotate pillar  2 to BIRD
Pillar 1 will also turn

Last, turn pillar 1 to WHALE
no other pillars will rotate

It should now look like this:

Saarthar Pillar Puzzle Solutions

Pull the lever.

The gate will open. Descend down the corridor. Tolfdir will show up.

To the north, there will be an alchemy lab that you can use.

Open the iron door to the WEST. You will reach a big chamber with some sort of magical green glowing sphere in the center.

Suddenly Jyrik Gauldurson rises from his throne! Attack him. But he seems invulnerable to your attacks! The source of his power must be the orb. Tolfdir now realizes this, and attacks the orb to drain Jyrik's power. On his mark, attack Jyrik with all you have got!

Once he is dead,  Loot him immediately to find a Gauldur Amulet Fragment and a writ of sealing. The Fragment increases your Magicka by 30 points so feel free to equip it. Reading the Writ of Sealing unlocks a new quest, forbidden knowledge.

Tolfdir will tell you that the Arch-Mage, Savos Aren, must be informed of the news immediately. He asks you to go deliver the message to the archmage. Your objective now is to talk to the Archmage in the college of Winterhold.

Before leaving the chamber, search the rest of the area for loot. There will be some gold in the urns on the wall. On the tomb, there will be the Staff of Jyrik Gauldurson which is pretty powerful and valuable. A burial urn will have a garnet in it.

When you are done looting, exit through the iron door to the west. Enter a grotto with blue glowing glyphs on the wall. Read the glyphs. You will learn the Word of power Ice, Ice Form. Loot the chest and leave.through the door to the Southwest. Activate the handle and exit through the gate. Work your way up the excavation and exit to Skyrim. Bring up the map and Fast Travel to the college of Winterhold. Find the Archmage, most likely in the Arch-Mage's quarters in the Hall of the Elements. Tell him you need to speak to him about Saarthal. Report your findings to him. Tell him about the orb you found. He will thank you for bringing it to his attention and will reward you with a Staff of Magelight.

This completes the College of Winterhold quest "Under Saarthal" and unlocks the third quest in the mage series, "Hitting the Books"