For the combat classes, the fastest way is to grind FATEs. While waiting for FATEs in each zone, you should ideally be in a group chaining together mobs. For FATEs with infinite monster spawn, you will instead be chaining mobs there for the chain xp gain. Should take you 4 days each.
For crafting classes and gathering classes, you will be turning in Leves. The fastest way is to simply buy HQ mats and items in the marketplace and turn them in for the Leves. Take care to mark the Leves that allow multiple turn-ins as those will allow you to gain up to 900% xp if you turn in all HQ items. With enough gil expenditure, it should take you 1/2 a day to 1 day for each class (assuming you have 100 leves). Otherwise, buy/gather the mats you need and try to craft HQ and turn them in. You get an additional 300% xp per craft/gather for doing this in addition to the leve. Cross-class skills are very useful and you will be required to learn how they work to reliably make money from crafting. Unfortunately, there is no set macro for crafting to just force yourself to HQ.
You will actually have to learn how to play the job. Keep in mind that you will have to run through all the dungeons and finish the Story to actually join and run end-game content. Try to learn how to play your favorite class then.
Also, as a warning, FATE grinding will not provide you with any meaningful amount of gil. This is especially true if you are chain-mobbing in between as the repair costs will easily overwhelm your gil generation and is not really sustainable. Consider doing class quests, side quests, and story quests for free gear upgrades and gil. They are relatively quick and easy if you do them in bulk sets while chaining or waiting for fates.
There is a severe difficulty spike at level 50, so try to put in the same amount of effort to run each dungeon with a class at least once. It might get redundant after the 5th class, but it will help you greatly at end-game.