It begisn when you meet Faralda on the bridge leading to the college. She will ask you, if the Dragonborn is not known for their spellcasting, to cast a particular spell on her. If the Dragonborn doesn't know that particular spell, tell her and she will sell you the tome for 30 gold.
Follow her across the bridge after that. On the far side you will be introduced to both Savos Aren, the Arch-Mage and Mirabelle Ervine. Mirabelle will offer to take you on a tour that leads to the Hall of Elements where a lesson is in progress. She will also show you the Hall of Attainment where you have a bed and desk to use. After that talk with
You will have to listen to a lecture by Tolfdir. He talks with the assembled students and tells that they need to be excited but they also need to show caution. He will ask the Dragonborn what they think on the matter.
He will ask you to cast a Lesser Ward spell and tests you. If the spell is unknown, then the spell is provided free of charge. He will then ask you to stand opposite him and he will throw a spell at you. The Lesser Ward will block the attack.
Once you cast the ward successfully the quest completes and you start the quest Under Saarthal.