Find the key to the top roof of the College of Winterhold

Places you can safely store your items include any houses you buy, your companion(s), and your horse.

Places considered to be owned by you (and therefore contain safe containers) are (spoiler warning):

• Arch-Mage Quarters (after you become Arch-Mage)
• The Dawnstar Dark Brotherhood Sanctuary (After you've finished the Dark Brotherhood quest line)

I have personally stored items in my apprentice quarters in the College of Winterhold without problems, however others have had stuff go missing from there, so be careful.

Other containers may not be safe. Given what some other answers have said, Gerder's house does not seem like a safe place for long-term storage.

Speak to Faralda at the entrance:

First, ask her why she's here.

Then ask her if you can enter the college.

When she asks why, your answer will choose which school of magic you join (destructive, etc)

She will ask you to perform a test with a certain spell. You probably don't have the spell yet, so buy it. It's going to be about 30 gold.

Once you buy the spell, if you are roughly level 3 or 4, you cast the spell at the seal on the floor. She will then lead you to the door of the college herself, no key required.