Young Wizard, the Spiral is all the worlds in Wizard101, Wizard City is the center of the Spiral.
If you wish to travel to other worlds in the Spiral, you will need to complete your Wizard City quests first. We can't just have students running all over the place until they've learned the ways of Magic.
Persevere through your training, and when the time is right, Merle will let you know that you are strong enough to adventure to other worlds. He will give you a Spiral Key, and from there you can access other worlds on the Spiral from inside Bartleby's trunk.
For a sneak peek at those other worlds, have a look here.