New main objective: Find the Arch-Mage
Get out of the Hall of the Elements and head towards the main square of the College of Winterhold. Tolfdir should approach you (screen above), informing you of the Arch-Mage's death and an attack on the nearby city. As you have probably guessed, you will be asked to help the citizens of Winterhold.
New main objective: Protect the town of Winterhold
In order to reach Winterhold, you will of course need to use the long bridge connecting it with the College (screen above). Be careful, as some parts of the bridge will be destroyed and therefore it will be quite easy to fall down. On your way there, speak with Faralda and ask her to join you in defending Winterhold. Interestingly, Arniel Gane will join the battle as well.
New main objective: Defeat the creatures infesting Winterhold
After you reach Winterhold it will turn out that it's under attack by a group of 10 Magic Anomalies (screen above). Start fighting them. As the Anomalies are very resistant, you can't hope to defeat them fast. Luckily however, they aren't resistant against any particular type of attack, so you will be able to fight them using any spell you like (or even a melee or ranged weapon).
Apart from your party member, Faralda and Arniel should also help you, thanks to which you won't have to rely solely on yourself. For the same reason I wouldn't recommend using AoE spells or Shouts, as you would unnecessarily hurt your allies. Keep fighting until you defeat all ten Anomalies. I'd recommend checking out the remains of each of them (screen above), as therefore you will gain a couple very good Soul Gems.
New main objective: Report to Mirabelle Ervine
Mirabelle didn't participate in the fight, so you need to return to the College of Winterhold, either by crossing the bridge again or using fast travel. Enter the Hall of the Elements and find Mirabelle (screen above), who will send you to the Labyrinthian to find the Staff of Magnus. She will also give you two items, the Torc of Labyrinthian needed to access the ruins and the very precious Savos Aren's Amulet.
College of Winterhold quest unlocked: Staff of Magnus