New main objective: Find Shalidor's writings
Note! This quest is optional and includes random objectives. I'd recommend repeating it from time to time, as you can therefore enrich yourself and at the same time "clear" some part of Skyrim from monsters. You will be able to take on this mission no sooner than after finishing the first main quest of College of Winterhold, i.e. First Lessons.
Find the mage known as Urag gro-Shub, who should be at the College's library, the Arcanaeum (accessible through the Hall of the Elements). During this conversation, ask him if he's searching for some special books by choosing the last dialogue option (screen above). Urag should mention an elven mage, a certain Shalidor and his writings that have been hidden all around Skyrim. The mage will then send you to find one of them, hidden in a random location.
Leave the College of Winterhold, open the world map and head to the destination chosen by the game. You unfortunately have to reckon that the area where the book is hidden will be guarded by enemies (screen above). During the fight, first deal with the enemies with magic skills and look out for elite enemies capable of dealing serious damage.
New main objective: Talk to Urag gro-Shub
Depending on the kind of area, Shalidor's writing might be hidden either on the surface or inside an underground location. The exact location can vary as well, as the book might be in a visible spot (like on a table) or hidden inside a chest (screen above). Having obtained the writings, return to the College of Winterhold and give them to Urag.
Don't worry that the quest ends, as Urag will need some time to transcript the contents of the book. I'd recommend returning to the mage after around two days (you can speed time up) and ask him about any progress (screen above) to receive the prize. Depending on your luck, you might receive Shalidor's Scrolls or a boost of one of your skills.