When you log in to Wizard101, you will see a screen like this:
Just click on the Wizard you would like to adventure with, and they will appear standing at the front of the classroom. Then click PLAY and you will log in as that Wizard.
The second Wizard will have the Dorm room, and cannot teleport to the first Wizard's castle. They can, however, purchase their own, or simply decorate their Dorm Room.
You are correct that all the characters on your account login have access to the Shared Bank in your Bank Storage. Whether Character 1 accesses it from their Castle or Character 2 accesses it from their Dorm, it is the same Shared Bank.