Best way to level from 52-60?

[Eul]

I always do dungeon when it's odd level number since it gives bonus chain. Also daily roulette helpful for fast exp.

[Canadane]

I'd always advocate dungeons over FATEs.
You get exp and experience, while a FATE will only give you one of those.

[MicahZerrshia]

Dungeons all the way (with supplementals thrown in between queues)

Honestly it's because it is always incredibly clear who just fate farmed their levels and never learned a thing about their class and rotations. Especially noticeable for tanks. I feel the fate-only mindset of leveling is why so many people come on these forums to complain about a job being bad or "too hard" (as if there is such a thing in this game) when many people are able to utilize the ability of the jobs just fine with little effort. You are actually making things harder on yourself and your pt members in the long run by solely fate farming exp, just to save an insignificant amount of time.

[AsahinaMyLove]

For easy goings:

I just did daily hunts for lving up! o___o
Did this around 10x on my 2 chars o;
If you really do ONLY hunts from lv51 on (because chocobo lv51 is need) It takes around 4-5weeks and you hit lv60 ;D
You can speed it up a bit by doing those weekly quests.

I did nothing else to lv up! o; To busy with work last months.
Burning away leves x10 was also a help.

[raela]

I do leveling/high/story/trial roulettes and hunts daily, then also grab the leve, fate, dungeon, and comm challenge log bonuses each week. For leves, now I'm doing 18 normal + 2 large leves a week. It gives enough variety that it doesn't feel totally miserable to me.

[Hurp]

As a tank I found that dungeons are easily the way to go, due to low que time. Though FATEs are arguably better if you can find a group for it, I personally have not seen any since I started with Heavensward content. The effectiveness of dungeon grinding(beside longer que time for dps) also comes with the party, what may have taken 20~min with one party, could end up taking 40~min with another. That's a huge difference in exp gain.

Also as MicahZerrshia mentions, it's a great way to learn your job and prepare for end-game!