https://www.reddit.com/r/ffxiv/comments/3egfp4/tldr_where_to_fate_grind_by_level/

1-10: Do your hunt log, job quests, and some random FATES in your hunt log zone.
10-20: Aleport (Western La Noscea)
20-30: Quarrymill (South Shroud)
30-35: Costa del Sol (Eastern La Noscea)
35-42: Camp Dragonhead (Coerthas Western Highlands)
42-50: Ceruleum Processing Plant (North Thanalan)
and then, as OP suggested:
50-52 - Coerthas Western Highlands
53-57 - Churning Mists
57-59 - Dravanian Hinterlands

FATE grinding is most effective if you have the Armory Bonus (a job two or more levels higher than the job you are currently leveling), and the ability to fly in the zone (Heavensward content).

Thanks for sharing! I don't think we have anyone that's brand brand new to the game, but in case you don't start in Limsa, swap out Aleport for East Shroud or Western Thanalan.