From what we have seen up to this point, you will need to level up to 10 on your Starting Class to be able to change Classes for the first time. You will also have to travel to the Guild of your new class and complete a Class Quest there. Sometimes this requires that you travel to another town to get to the Guild you want to join. The Class Guilds are in the respective cites as listed above.
After you complete the Class Quest, you can now switch between any of your unlocked classes simply by changing weapons.
EDIT: Below is a list of Advanced Jobs and which of the Base Classes you would start as. This is not something that you will need to be concerned with until you have at least reached Level 30 in your Base Class.
Maurader ---> Warrior
Arcanist ----> Summoner & Scholar
Gladiator ----> Paladin
Pugilist -----> Monk
Thaumaturge ----> Black Mage
Lancer -----> Dragoon
Archer -----> Bard
Conjurer ------> White Mage
EDIT: While gathering this info I found this Starter's Guide and it is awesome!
Edited, Aug 15th 2013 7:08pm by Gnu