Throughout your adventures, if you asked shopkeepers and merchants where to get new spells, they probably recommended that you go to the College of Winterhold. If you are a spellcaster, you want to go there early in the game to open up some new horizons in magic. If you are more of the warrior type,  you can delay going there, as you will be sent there eventually anyway during the main campaign quest: "Elder Knowledge"

The fastest way to get to the College of Winterhold is to travel to Whiterun, and from there hire the carriage thats outside the city to take you to Winterhold.

When you arrive at the city of Winterhold, head North. Climb up to the bridge. Faralda, a Senior Wizard from the college, will be there and tell you to cross the bridge at your own peril. The way is dangerous and the gate will not open. You shall not gain entry. Ask her what is this place. She will tell you that it is the college of winterhold, a safe haven for the mages in Skyrim, a place of wisdom and arcane knowledge. Ask her why she is there. she will tell you she is there to assist those seeking the wisdom of the college. She will ask you why you are here. Ask her if you may enter the college. Tell her why. She will give you a entry test, to cast a spell and show that you know magic. If you don't have the spell you will be able to buy it from her for cheap, so you might as well since you will be paying less then if you were to buy the book from a shop

The spell she will test you on can either be Firebolt (Destruction), Flame Atronach (Conjuration) or Mage Light (Alteration)

If you check your quest journal it should say:
First Lessons
In order to join the College of Winterhold, I  need to show I'm capable of simple magic by casting a firebolt spell.

Cast a firebolt spell.

Step on the circular platform in front of her and cast the spell onto the seal on the floor. If casting the firebolt, make sure NOT to cast it on her as it will be considered an attack.

TIP: If you are having problems with casting the spell she wants, try a Dragon Shout instead if you know one. She will be impressed and realize you are a Dragonborn and worthy of entering the School of Magic.

She will congratulate you and welcome you the the college. She will lead you across the bridge. She will tell you that once inside, you will want to report to Mirabelle Ervin, the master wizard. Follow Faralda across the bridge, then talk to Mirabelle who is standing outside the door to the college. Tell her you were told to come see her. She will give you some robes and hoods that you can wear, then tells you she will give you a tour before you attend your first class. Tell her you would love to have a look around, then follow her. Don't wander off.

If you look at your quest journal, it will now say:
I have joined the college of Winterhold. Mirabelle Ervine has agreed to show me around in order to become familiar with the grounds.

Your objective is to tour the College of Winterhold.

Take the tour. After the tour is over, you need to find Tolfdir, who looks after the new Apprentices, and listen to his instructions. Enter the Hall of Elements that's in front of you. Go past the gate and find Tolfdir who is talking to a group of apprentices. He will give a lecture about magic. The apprentices are impatient and want to start with something practical right away. Tofdir will turn to you and ask you what you want to do. Tell him you want to learn something practical. He will teach you a ward. He will ask you if you are familiar with Ward spells. If you do not know any, he can teach you a basic one. Follow his instructions.

Stand on the platform across him. Open your magic menu and locate the Lesser Ward spell. Equip it, and cast it. Tolfdir will cast a firebolt at you, which will be absorbed by the Ward.

This completes the first College of Winterhold quest, "First Lessons", and unlocks the next quest, "Under Saarthal".