Ok, here's how it works:

You start out as your beginning class. You can not change that class until you complete your class's level 10 quest. After that, you are able to travel to the other classes' guilds and join those classes. At that point, you will be able to become that class just by changing weapons.

For jobs, you have to level a job's primary class (Thaumaturge in the case of the Black Mage) to 30, complete that level 30 class quest, and then level the job's secondary class (Archer for Black Mage) to 15. At this point, you will travel to the primary class's guild (Thaumaturge guild) and they will give you a quest. When you finish that quest, you will receive an equippable soul crystal that will make you a specific job, in your case, a Black Mage.