The fastest way to level is getting into a party.
With your party you can:
-do leves
-do dungeons
-do FATEs
-kill monsters quickly

The best way to max your EXP/time is to get into a FATE party and kill monsters during any downtime (when you aren't doing FATEs) this'll net you way more EXP than sitting next to the nearest aetherlyte doing nothing. BUT THERE IS MORE!
Another neat thing about FATEs is that there are different kinds, collection and boss FATEs are the BEST kind. Why is that?
FATEs spawn adds at a very fast pace and it just so happens that these sorts of FATEs don't end by killing the rapidly spawning mobs. So what do you do? Simply kill the mobs! Do NOT complete the FATE's objective until the last few minutes.
A REALLY popular example of this is the Dark Devices (commonly known as DD) FATE chain where people will generally camp the mobs rather than finishing the FATE for the big EXP at the end.