Welcome to Eorzea!

As Sagittarian stated, you do need to go to Gridania to unlock Archer, but you don't have to wait until you unlock the airship as you can simply travel there by foot or chocobo porter (It'll just take longer). To answer your question, when you meet the requirements for Black Mage (lvl30 Thaumaturge and lvl15 Archer) you would go to the Thaumaturge's Guild to find the quest to become a Black Mage. Make sure you finish the class quests though because if you did not do them, you would have to do them in order to unlock the quest for Black Mage. Once you finish the quest for Black Mage, on your Thaumaturge, simply equip the Soul of the Black Mage stone you get from the quest and you're a Black Mage!!! Have fun blowing stuff up and welcome to the dark side.