You will find out that the Mage Guild quests are quite long and that joining the college is nothing like entering Hogwart. You will have to complete multiple dungeons and dive deep into the mystery before that achievement unlocks.
First Lessons*** Spoiler - click to reveal ***After Faralda grants you access to the College, you will have to talk to the Wizard Master Mirabelle Ervine. She will take you on a small tour of the premises and leave you to learn with Tolfdir. The lesson will give you the Lesser Ward spell if you don't have it. Following the demonstration, Tolfdir will take everyone on a field trip, in the dig site of Saarthal.
Under Saarthal*** Spoiler - click to reveal ***Meet with the group outside of Saarthal. You will have to explore the archeologic site until you find Arniel Gane. He will ask of you to collect a few artifacts around.One of them, the Saarthal Amulet, will trap you when you take it. You have to equip it and destroy the wall behind the tablet with a spell or a shout. This opens access to a new aera of the mine, where you will face Draugr with Tolfdir at your side. There will also be a lot of traps set out. You will eventually make it to a puzzle door with six pillars; the combination is actually drawn over each pillar, use Illumination if you have trouble discerning the glyphs. Keep pushing forward and you will be at another puzzle door with four pillars. Here the solution is obvious but the twist is that some pillars activate others. So you have to activate the far left one first, then the close left one, then the far right one and finally the close right one.
You will eventually make it to the sphere room (the one and only: Eye of Magnus!) where a boss battle awaits you. Jyrik Gauldurson will be summoned from the orb and will be invincible at first. When Toldfir weakens the sphere, Jyrik is open for damage. Be warned he is immune to all destruction magic, so taking him down from up the balcony with some arrows is a sound solution. You will then be awarded a good deal of magical staffs and a fragment of the Gauldur Amulet as loot. Talk to Toldfir to continue the quest and be sure to get the word of power for Ice Form before you leave. Then meet with Arch-Mage Savos Aren at Winterhold to conclude this quest.
Hitting the Books*** Spoiler - click to reveal ***Talk to Urag Gro-Shub about the connection between Saarthal and the Pijik Order. He will tell you the documents that hold the information you're looking for have been stolen by Orthorn, who fled to Fellglow Keep. So you head to the dungeon and start clearing the place of all type of mages, until you make it to Orthorn's cell. He will tell you that The Caller has what you are looking for. You can either leave Orthorn in jail or free him, have him follow you or be sent on his way. Clear the second part of the dungeon and you will meet The Caller at the end. Your objective here to get the books, either by dialog, theft or violence. Just know you can hand in Orthorn (if he's with you) for the books and that if you choose to fight, you're in for a brawl with a strong conjurer that can use restoration and teleport around. Once you get the books, head back to the Arcanaeum.
Good Intentions*** Spoiler - click to reveal ***Meet with Tolfdir in the Hall of Elements, where the Eye of Magnus has been placed for observation. Discuss the artifact and Ancano will come for you to announce you have visit from someone of the Psijik Order. You will all meet in Savos Aren's quarters and Quaranir will reveal you must seek the Augur of Dunlain. Talk around about it, but only Tolfdir will reveal you its location, which is The Midden, that lies beneath the College. You will fight some Draugr and Atronachs there, untill you reach the Augur. Converse with it and you will be on the quest to retrieve the Staff of Magnus. You have to talk to Savos Aren (in the Hall of Elements) and then to Mirabelle Ervine to go on.
Revealing the Unseen*** Spoiler - click to reveal ***Talk to Mirabelle to learn some members of the Synod came asking about the Staff, but left empty handed toward Mzulft. Go to the mark she inscribes on your map and enter the Dwemer ruins. You will find Gavros Plinius of the synod deadly wounded right at the entrance. Read his log and you'll find out you need to find the crystal he was carrying and head to the Oculory. Into the ruins you will encounter Dwemer machinery to fight, but after a while you'll principally face Falmers and some Chaurus. Keep going until you reach the Boilery and then the Aedrome. In its first major room, you will find some enemies, one ofthem carrying the crystal. Slaughter everyone and retrieve the item. You will then enter another big room with a door on each side. The one on your left leads to the Oculory, but you'll need the key that is to be found through the door on the right; the key is in a big room in which you have to fight a Dwemer Centurion. When ready, go to the Oculory. Paratus Decimius is waiting behind the door.
He was waiting for Gavros, but you'll have to tell him the bad news. Talk to him a while and he'll take you to the Armillary Sphere. You have to insert the crystal in its middle and perhaps wait for daylight so a light beam strikes the lens. You then have to use the Frostbite spell (Flames to go backwards) to adjust the crystal so the three beams emanating from it each light a different rotating ring of the ceiling. Then go to the console atop the ramp and press the pedestal buttons until the green spheres of each rotating ring is stroke by a beam; each button controls a different ring, so you should have no trouble. Succeeding will result in a map of northern Tamriel being projected on the wall. Talk to Paratus until he reveals you the staff is in Labyrinthian.
On your way to the exit, a representative of the Psijik will appear to you. He'll insist you return to Winterhold fast. Once there, enter the college and you'll see Ancano has been trifling with the Eye and the others are trying to stop him. Follow the dialogs until the screen flashes blank. That concludes the first part of this questline.