Much like reading a chapter book, Wizard101 is sequential game play (before you move to level 2 you must complete level 1).
Even if you have purchased a Membership you must first complete all of the quests in one world to be able to move on to the next.
The same thing goes for Access to some of the streets or areas within a world. For instance, when you've successfully completed the quests for Wizard City you will be given a quest to enter into the next world, Krokotopia. Once you have your own access there, as long as you have a Membership or have purchased the Premium zone with Crowns, you will be able to enter that world freely through the Spiral door.