Once you complete the level 10 class quest for your starting class, that unlocks the ability to add other classes. To add another class, go to that class's guild and take the "So you want to be a <class name>" quest from the guild's receptionist. It's a quick quest that just has you go talk to the guildmaster and he/she gives you a starting weapon or tool for the class.

Once you've unlocked a class, switching to that class is just a matter of equipping the type of main hand weapon or tool used by that class, so anytime you're holding a thaumaturge wand in your main hand you're a thaumaturge, anytime you're holding a sword you're a gladiator, anytime you're holding a bow you're an archer, etc. (You'll probably want to set up gear sets for each class. That lets you switch all your gear at once to a set appropriate to the type and level of that class.)

There are no limitations on how many classes/jobs you can unlock and advance. You can do all 19 classes and all 9 jobs all the way to level 50 if you want. But each class starts at level 1 and has to be advanced separately from all the other classes. Jobs are different in that they don't actually have a level of their own, but use the level of their base class.