Once you've demonstrated your basic magical capability to Faralda, she'll agree to let you in, and tell you to speak to Mirabelle Ervine, the woman who basically operates the College of Winterhold as a whole- while Savos Aren may be the Arch-Mage and headmaster, it is Mirabelle who does most of the work.
Faralda then heads slowly across the bridge leading to the College, using her magic to light up the two glowing pools along the way as a signal to those inside the building itself. If you wish to gain some training in Destruction magic, you can ask Faralda now- or later on whenever you pass through this area.
With the First Lessons quest now active, continue into the building and seek out Mirabelle Irvine. The gate to the main courtyard should be open, so enter and carefully round the glowing beacon pool in the center, and pass the statue of a mage that overlooks it. You will find Mirabelle in a rather uncomfortable discussion with Ancano, the Thalmor embassador to the college. When they are done, get Mirabelle's attention and let her know you were told to speak with her.
Mirabelle will give you a set of mage's clothing- a Novice Robes of Destruction, a Novice Hood of Magicka, and a pair of plain boots. Wear them or not, whatever your preference, but she offers to guide you in an introduction to the school. You can ask for a delay, but your better option is to agree.
She explains the basic layout of the central area of the building, with the Hall of Elements, Arch-Mage's Quarters, and the Library, then indicates that she will show you the main living area. As you follow her, she explains that the locals have something of a dim view of the College- this is why you had to get past Faralda to enter, as they are wary of any potential incidents. Mirabelle says she's not anticipating an attack or anything on that level, but prefers to be cautious just in case.
Mirabelle guides you to the Hall of Attainment, the novices' quarters. She assigns you a dormitory with a bed and a desk (and a soul gem) and gives you free rein in the area, so long as you don't bother the other students. This does mean that you are free to gather the vast majority of the materials not only in your room but also in the two general-use rooms on this floor of the tower.
Once the basic rules are laid out, she turns to guide you back to the Hall of the Elements to meet the elder wizard Tolfdir for an introductory class. Mirabelle stops by the central beacon pool, leaving you to enter on your own.
