You start out leveling 'classes' (ie. Conjurer, Gladiator, Archer to name a few). Most jobs (ie. Paladin, White Mage, Bard again to name a few) are unlocked by leveling a main job to level 30 and a secondary job to level 15. For White Mage, you need to level Conjurer to 30 & Arcanist to 15 and then complete a quest which gives you a soul stone to equip to become a WHM.
This is true for all jobs currently except Rogue to Ninja which doesn't require a secondary job (and neither will the new jobs coming in the Heavensward expansion).

Using WHM again as an example, once you unlock the job it will be at the level of the main class for that job. Meaning that if your CNJ is level 30 when you unlock WHM, your WHM will start at level 30 and both WHM & CNJ will level together. Although once you unlock the job (WHM) there is very little reason to use the class (CNJ).

As for your quest question, it's entirely up to you. Some people use all the quests they can find to level their first character and others will save some quests for any other classes/jobs they level.
It is relatively easy to level a job to 50 with a combination of quests, fates, dungeons, leves & hunting logs without using up all the quests available.

Welcome to Eorzea and I hope you enjoy the game