Speak to Faralda and tell her you want to join the College. She'll tell you to cast a random spell from one of the five schools of magic. If you don't have the spell she asks for, tell her and she'll sell it for 30 Gold. Now the new objective is to cast your spell--equip the spell and cast it. Alternatively, you can pass a persuasion check to avoid the test, OR say you are Dragonborn if you've complete Dragon Rising from the main questline. If you say you are Dragonborn, Faralda will ask you to use a Shout. Afterwards, whatever method you use, you'll be granted access to the College. Follow Faralda all the way to the entrance of the College and once inside, you'll see Mirabelle Ervine speaking with someone.
Note: Faralda will ask you to cast one of the five following spells: Firebolt (Destruction), Conjure Flame Antronach (Conjuration), Fear (Illusion), Healing Hands (Restoration), or Mage Light (Alteration).
When she's free, speak to her. She'll give you the Novice Robes of Destruction , which are pretty nice for a destruction mage (you have to start somewhere!). She will now offer you a tour, so follow her into the Hall of Attainment.
She'll show you your room, which you can use when you want to. There's a lot of useful items in the other rooms in this area, so if you're into stealing you can likely fill your pockets with some items of worth.
Return to the main area and go north to the Hall of the Elements. In here, open the gate in front of you and listen to what Tolfdir has to say. During the lecture, you'll receive the 'Lesser Ward ' spell, as Tolfdir wants to demonstrate it to the class, so equip it and keep using it until Tolfdir fires a spell on you. Afterwards, you'll complete the first quest of the College.
Note: Want to level up Illusion in a hurry? Find Drevis Neloren at the college and buy the spell Muffle from him. You can cast it while you travel around in your spare time and it will level up your skill in Illusion fast. Be careful not to put all your eggs in one basket though!
Join the College of Winterhold