To join the college, head to Winterhold and talk to Faralda, the women in front of the bridge to the college. Depending on what you say, you will either have to perform a Dragon Shout or cast a certain spell (which you can buy) to be admitted to the college. After you get in, you will have to speak to Mirabelle who will give you a tour.
After she gives you a tour and shows you to your quarters, you will be asked to join a lecture. Go to the lecture hall and listen to Tolfdir. You will be asked to perform a spell (you will learn it for free if you don't already know it.) and he will attack you. After this, the quest will complete.