Leveling up DPS/Healer/Tank

So you decided to level up a new class, but wait there are no mainstory quests for you to do anymore. Well don’t start to panic just yet, leveling up a different class than the one you started with isn’t hard at all.

Here are some more tips for leveling up:

Whenever you eat some food (in the game of course) u gain a small exp bonus, i recommend to always have this bonus up whenever you are leveling.
Queue up for dungeons whilst doing fates/levequests, dungeons give some serious exp and you will learn how to play your new class better.
Make use of your chocobo, if you are not queuing up for a dungeon always have your chocobo summoned, it makes doing fates/levequests much easier.
Join a fate party, if there is a fate party available i recommend joining it, not only will you clear fates faster you will also gain more exp because your party members will trigger the kill combo bonus.
Don’t forget to do your class quest every 5 levels or every 2 levels once you start with Heavensward.

The following guide from Reddit shows the most efficient way to level a class from level 1:

Level 1 - 8 : Hunt log
Level 8 - 10: Random fates in a starter area.
Level 10 - 13: Hunt Log
Level 13 - 20: Fates in Aleport
Level 20 - 22: Hunt Log (Ask a friend or stranger for help to finish it.)
Level 22 - 28: Leves in Quarrymill, Two in particular. A level 20 one that involves collecting traps and a level 25 one that involves collecting water. Neither require killing so you can turn it up to max level.
Level 28 - 34: Leves in Costa Del Sol. There are three easy Leves that are faster and offer more XP than fates. Out to Sea, which involves click 5 items, no combat. Necrolopis, which you can do with only having to kill one monster and Culling the Herd which involves killing five monsters. The goal is to do Out to sea as much as you can, if its not available, do Necolopis, if that's not available do Culling the herd as a last resort. If Cancer spawns, join a fate group and do that.
Level 34 - 42: Fates in Central Coerthas Highlands
Level 42 - 50: Fates in North Thanalan. I recommend waiting until level 42 before heading here.
Level 44 - 48: DD first room runs if you can create a premade party. Ideal setup: Dark Knight, Astrologian, Summoner, Machinist.
Level 50 - 55: Fates in Western Coerthas Highlands.
Level 53 - 55: Sohm Al
Level 55 - 58: Fates in Churning Mist.
Level 55 - 57: The Aery
Level 58 - 60: Fates in Dravanian Hinterlands
Level 57 - 59: The Vault
Level 59 - 60: Great Gubal Library
Challenge Log: Be sure to unlock and do the challenge log each week. Its essentially a free level or two. The challenges experience reward scale on your level, so its best to use them 30+ in my opinion.
On a side note. I highly recommend using dungeons to level 53+ and doing an occasional dungeon or two every 5 or 10 levels so you have an idea on how to play your class. I can't stand it when I get someone in a party post level 40 who has no idea how to play their class well.

Guide Source: Guide to Leveling