Activating the quest
Activating the first quest of College of Winterhold first requires you to find the location. Head to Winterhold, located in the north-east part of Skyrim (screen above). If you have already visited the town, use fast travel of course. Otherwise, you will have to plan your journey. The easiest way of reaching Winterhold is hiring a carriage (e.g. in Whiterun Stables ).
After reaching Winterhold, look for the large bridge (screen above), as it's the only way of getting to the College of Winterhold.
As you try to cross the bridge, you will be stopped by Faralda, who will tell you that only authorized people can enter the College. The easiest way of solving this problem is accepting Feralda's proposition of a short test that could prove your usefulness and magic abilities (screen above).
Faralda will ask you to cast a random spell. If you know it, you will be able to skip to the next step. Otherwise, you will need to unlock it and the easiest way of doing it is by asking Faralda for help (screen above), as learning it will cost you only 30 gold.
Regardless of whether you had the spell or had to ask Feralda to teach you it, you will now have to cast it. Choose the spell and set it as active. In most cases you will need to cast it on the seal on the ground (screen above), except if you're asked, for example, to perform a summoning, in which case you should aim somewhere beside you. Remember not to aim at Faralda, as it might provoke her to a fight!
There are also two other methods of gaining access to the College. If your character has highly developed Speech, you can try persuasion. Another solution is informing Faralda that she's speaking with the Dragonborn (screen above). For obvious reasons, you will be able to use that option only if you have unlocked at least on Shout (by meeting the Greybeards during the main quests).
Faralda will want you to prove to her that you're the Dragonborn. As a result, press TAB, open the Magic menu and activate any you like. I'd recommend choosing Unrelenting Force and pressing Z (screen above), thanks to which you will only slightly hurt her.
Regardless of which method you choose, Feralda will eventually let you enter the College, taking you to the main square and sending you to speak with Mirabelle Ervine. Note: If you intend to take care of the College quests after finishing the main story, you should remember that you gain access to it during Elder Knowledge.
College of Winterhold quest unlocked: First Lessons
New main objective: Report to Mirabelle Ervine
You should find Mirabelle Ervine on the main square of College of Winterhold (screen above). Speak with her and she will give you the basic mage equipment, namely Boots, Apprentice Robes of Destruction and Apprentice Hood of Magicka. Using them is completely optional, so feel free to use better mage equipment if you have one. In the end Mirabelle will offer you a tour around the College, which you should agree to.
New main objective: Tour the College of Winterhold
Head after Mirabelle. The woman will take you to the Hall of Attainment, showing you your assigned room (screen above). Wait until you return onto the main square to learn that the mage Tolfdir will conduct your first training.
New main objective: Listen to Tolfdir
Use the door leading to the Hall of the Elements (screen above). After getting there, open the metal gate in front of you to reach the area where Tolfdir is giving a lecture. You will have to answer a question concerning the danger of using complex spells, though the answer won't really matter.
Tolfdir will soon inform the gathered students that the first test will concern using protective spells. The mage will once again turn towards you, asking if you know any spells of that kind. If you don't, you will automatically learn the Lesser Ward from the Restoration group. In case you already know a spell, you will be able to choose between learning more about it and skipping to using it.
Tolfdir will want you to cast the spell, so start off by choosing it from the list as setting as active (you can assign it to any hand). Afterwards stand on the marked plate (screen above).
Turn around towards Tolfdir and hold down the mouse button to which you assigned the spell. Don' let go of the button and note that the spell protected you from the fireball that Tolfdir threw at you (screen above). Now just listen to what the mage has to say. He will invite the students for another lesson which will this time take place outside the College, in the Saarthal ruins excavation.
College of Winterhold quest unlocked: Under Saarthal