Very New

[Frizze]

Not mentioned above, but if you have interest in crafting or gathering it works in the same way. You unlock changing classes by doing the level 10 quest of the class you started as, then you go to the guild youre interested in joining. Once you start unlocking multiple jobs, its highly advised that you set up gear sets for each. On the character window above the picture of your character and all your gear you press the button that looks like a single sheet of paper to open the gear set tab, then at the top of that tab you press the + button to create a new one for the class you currently have equipped. Then switch to the next class, put on all the gear you want to wear, and hit that + button again. This allows you to switch all your gear at once. Anytime you get an upgrade to your current set, you can press the button of 2 arrows switching positions(above your character, next to the gearset tab) and it will update your set to what you now have on(this also works if you keep the same gear, but glamour over it so it looks different).

As for crafting/gathering, Ul'Dah has:
Alchemist(crafter who makes wands and potions)
Goldsmith(crafter who makes staves and jewelry)
Miner(gatherer who collects soft and hard metals, along with rocks)
Weaver(crafter who makes cloth-based clothing/gear)

Limsa has:
Armorsmith(crafter who makes metal based clothing/gear)
Blacksmith(crafter who makes weapons and tools)
Culinarian(crafter who makes food)
Fisher(gatherer who collects fish)

And Gridania has:
Botanist(gatherer who collects wood and plant based materials)
Carpenter(crafter who makes spears and bows)
Leatherworker(crafter who makes leather based clothing/gear)

As for your other questions, continue following the main story questline. Around level 20 it will give you your "first" mount(it is possible you could have one before then from a couple sources, but they wont be usable until you do this), and unlock your ability to use/get more of them. Then around 30 youll have the option to unlock a companion. Just make sure you keep up with the main questline.