Each class has its own level, and whenever you change to that class you'll become the level you've earned on that class. (The character window has a "Classes" tab that shows your current level in each class in the game.) Apart from the class you begin the game with, any other classes have to be unlocked by doing a simple quest at that class's guildhall. (Ones not yet unlocked are level 0.) After that, you level each class independently of the others, with whatever Experience you earn going towards leveling whichever class you're on at the time. (There are a few exceptions to this if you get into crafting and gathering. For instance, Supply and Provisioning missions always give their XP to the class they're designed for, regardless of which you are when you turn them in.)

Once unlocked, you can alternate between classes by just changing your main hand weapon or tool, but more commonly you'll want to change all your gear at once to whatever you have that's best suited to a given class at its current level. For this, you assign gear into Gear Sets. In the character window, just above your character portrait and equipped gear, there's a button with a page icon for bringing up your gear set list. You can create new gear sets or switch between gear sets from there.

Once you get to the point of unlocking jobs, those work a little different than classes. Each job is considered a special type of their base class. So a summoner, for instance, is a special type of arcanist and will be whatever level you are as arcanist. Experience points earned as either an arcanist, summoner, or scholar are all arcanist experience. It's always the class that levels up and the job(s) that go with that class share its level. (At least this is how all the jobs currently in the game work. The new ones coming with the Heavensward expansion are a bit different because they don't have base classes. I'm not sure on the details yet for those.)