"Jobs" (most of them anyway) are the advanced version of classes. You'll need to get one class up to level 30 and another up to level 15 in order to unlock the job associated with the level 30 class. In the case of Gladiator, that can be advanced to Paladin once you have Gladiator level 30 and Conjurer level 15.
Gladiator (+Conjurer 15) -> Paladin
Marauder (+Gladiator 15) -> Warrior
Lancer (+Marauder 15) -> Dragoon
Pugilist (+Lancer 15) -> Monk
Rogue (+Pugilist 15) -> Ninja
Archer (+Pugilist 15) -> Bard
Thaumaturge (+Archer 15) -> Black Mage
Conjurer (+Arcanist 15) -> White Mage
Arcanist (+Thaumaturge 15) -> Summoner
Arcanist (+Conjurer 15) -> Scholar
(And no, it's not a typo that arcanist appears twice in that list. That class gets two jobs that it can advance to, while all the others just have one.)
Then the extra jobs added with the Heavensward expansion (Dark Knight, Astrologian, and Machinist) are a bit different. Those require actually reaching the Heavensward area, Ishgard, though, before you can unlock them. That won't be until level 50 at least, so you won't be getting to them for quite a while yet.
No. One of the cool things about this game is that a single character can have multiple classes and jobs. You can advance all the classes and jobs in the game all the way to endgame if you want to, or any combination of them that you're interested in. As soon as your first starting class reaches level 10 and does the level 10 quest from your class's guildmaster, it will unlock the ability to add other classes as well. Then you go to the guild of each other class you want to add and do a quest there to unlock it.
If you're interested in crafting and/or gathering, this is also the point at which those can be started. Crafting and gathering classes are added just like additional combat classes are. It's available once you reach 10 on your starting class. (It's only combat classes that upgrade to jobs, though. Crafting classes remain as a class all the way through the game.)