The main idea is to stick around FATEs either higher level than you - or simply a few levels lower. For a general rule of thumb: If you have to sync down for more than half the FATEs in your current zone, it's probably best to move on.

Also a note, if this happens to be your first class you're leveling up: DO YOUR STORY QUESTS! These are indicated by the flaming sort of quest symbol next to them. These unlock things such as Grand Companies (which in turn unlocks your Chocobo), gives access to most dungeons and all Primal fights, and Retainers (personal banks, and selling mules).

1-10: Hunting Log + Class quests
Not too much here, generally much faster to go along with the Hunting log, rather than waiting for FATEs to pop - though mixing them in certainly makes it faster.

10-20: Western La Noscea - Aleport
Sticking around Alepoort, and east to the ones by Swiftperch

20-28: South Shroud - Quarrymill
Any FATEs in the northern part of the zone, by Quarrymill and west by Buscarron's Druthers.

28-34: Eastern La Noscea - Costa del Sol
Stick mainly to the east side of the map, as it's where the majority of parties stay.

34-40: Coerthas Central Highlands - Camp Dragonhead
More or less just follow your party around; you'll mainly stick around Dragonhead, going to whatever pops first, and Svara's or Eyes above any other FATE.

40-50: Northern Thanalan - Ceruleum Processing Plant
(I believe) You'll be going for every FATE in the zone - though "Dark Devices will be the main focus, making this zone a bit better than Mor Dhona. In Dark Devices, don't attempt to complete it by killing the Lambs to advance the FATE, instead, AoE farm the near-instant respawns. Overall, the entire chain can earn between 100-200k xp if you have a party good at claiming. The only downside is your durability dying quickly.

Helpful tips for FATE farming
1).Try to join a Grand Company before farming FATEs. When part of a Grand Company, you earn seals upon completing every FATE, which can be exchanged for fabulous prizes(!) like gear, consumables, and fun Chocobo armor. Additionally, joining a Grand Company means you can unlock your Chocobo - which makes the arduous task of running between each and every FATE a less annoying task.

2).You don't need to stay for a FATE to be completed to receive your xp. You can get enough credit for gold, then just run off to the next one while the first one is still being finished.

3).If doing a hand-in FATE, all you ever need to turn in for gold is 6 items.

4).Spamming point-blank AoE's (Flash, Arm of the Destroyer, Blizzard II) are generally the best way to get the most claims.