Welcome to Eorzea!

There are a ton of "how do I level" posts but the short version is:
  • First class - just do the quests available at least through level 10. Make sure to do the level 5 and 10 class quests!
  • Other classes and first class after level 10 - the hunting log and challenge log both give good xp boosts. The hunting log entries can only be completed once for each class. The challenge log resets every week.
  • In general - leves, leves, and more leves. Particularly for crafting and gathering.
  • Don't neglect guildhests - you get bonus xp for doing each one for the first time with each class
  • After level 15 - dungeons and some FATE grinding
  • After 2+ dungeons unlocked - the low level roulette gives a lot of bonus xp so do it daily

You will also accumulate bonus rest xp while you are in a sanctuary or inn, even when you are offline. You will notice is as a change in the color of the xp bar.
It is very easy to burn through all of the bonus xp and other "low hanging fruit" if you are trying to powerlevel but the game basically hands you at least a level a day. Doing just the low-level and guildhest roulettes each day gives me ~1.5 levels.