For levels 1-15 you can power level with a high ilv friend. Find a good spot with a LOT of mobs, tag them one at a time, and have your friend kill 'em quick. Best to find a long stretch of them or a cycle of them, and keep the exp chain bonus going! You cannot be in the same party as said friend though. If you're also both willing to stick it out for a bit longer, this method stays effective until lv20ish depending on the mob's level.

PotD isn't super efficient until lvs 30+ imo, but i haven't done the math or anything. Remember that for tanks and healers though, assuming queue times are short, regular dungeons are still supposed to be faster in addition to the fact that rested, food, and fc bonuses are applicable.

Leves can be even faster than PotD if you have a willing lv60 friend. Get in the same party and have them rip everything apart for the leve(careful for the /soothe ones!). Turnins are as quick as a minute. The trick for leve repetition is to accept about 3 of them with intent on only doing 1 or 2.
Example: in Whitebrim Front you would want to spam "This Area is a Bit Sketchy" and "Necrologos" since they only involve killing target monsters. Accept these 2+1more, do one leve, and turn it in. Hopefully, the one you just did will reappear in the list. If not, do the other and upon turnin one/both will reappear in the list. Occasionally, none will come back, and you'll have to do the 3rd one to get back into the cycle.

Fates are good for the lv20 gap, especially if you want the seals or leveling the 3 GCs. Still not the fastest choice.