Bolded areas are recommended for that range.

-You do not need to stick around for the entire FATE.
-If you feel comfortable that you have done enough to receive full exp for the fate, you can leave early to join another FATE that may have popped.
-This will help you optimize maximum exp potential.
-Make sure to join a party. Often if you shout in an area that you are "looking for group" someone will pick you up.
-Being in a party not only allows you to receive bonuses to parameters, it allows you to share everyone's exp they are getting from monsters begin killed.
-There is also a group contribution feature so it is easier to get a gold medal.

*WESTERN THANALAN*: Scorpion Crossing (3 fates)
CENTRAL THANALAN: Southwest of Black Brush Station (4-5 fates)
LOWER LA NOSCEA: South of Red Rooster Stead (3-4 fates)
MIDDLE LA NOSCEA: South of Summerford Farms (3-4 fates)
CENTRAL SHROUD : Zone out of Gridania (3-4 fates)
NORTH SHROUD : Zone out of Gridania (2-3 fates)

LEVEL 5-10
*WESTERN THANALAN*: West of Scorpion Crossing (5-8 fates)
CENTRAL THANALAN: Black Brush Station/North (4-6 fates)
LOWER LA NOSCEA: North of Red Rooster Stead (4-5 fates)
MIDDLE LA NOSCEA: West of Summerford Farms (4-6 fates)
CENTRAL SHROUD: Bannock (4-5 fates)
NORTH SHROUD: South of Hyrstmill (4-6 fates)

LEVEL 10-15
*WESTERN THANALAN*: Vesper Bay/Horizon (4-6 fates)
-This whole area is great, the Horizon side is lower level you could actually hit there around 7 or 8. The Vesper Bay side is higher level, good place to FATE grind with a group.
-"Bubble Trouble" lv.13: Vesper side has a Bubbly Bernie crab fight that is pretty good. Reference to a FF11 NM.
CENTRAL THANALAN: East of Black Brush (3-4 fates)
LOWER LA NOSCEA: Moraby Dry Docks (4 fates)
MIDDLE LA NOSCEA: North of Summerford (5 fates)
WESTERN LA NOSCEA: Swiftperch (5-6 fates)
CENTRAL SHROUD: North of Bannock/S of Bent Branch (5-7
EAST SHROUD: West of Hawthorne Hut (3-4 fates)

LEVEL 15-20
EASTERN THANALAN: Camp Drybone Area (7-9 fates)
*WESTERN LA NOSCEA*: Aleport (6-7 fates)
-Should be noted Skull Valley is loaded with tons of fates spanning lvs 15-19. Many regard this as the best FATE grind for lvs 13-20.
EAST SHROUD: East of Hawthorne Hut (5-6 fates)

LEVEL 20-25
WESTERN THANALAN: North of Vesper Bay (5 fates)
UPPER LA NOSCEA: (1-2 fates)
-"Simurgh is the Word" lv.23: Surprisingly quick fight against a planty bird like dragon. Pretty straight forward fight watch AoE's, Courel pups spawn near him take care. Decent exp.
EAST SHROUD: North of Hawthorne Hut (2-3 fates)
-"Lazy for You" lv.20: Fight Lazy Laurence a giant ochu. Watch out for his minions as they multiply quickly. Battle takes awhile but great exp reward for the level. You receive a mini fungus minion as a reward as well.
*SOUTH SHROUD*: East and West of Quarrymill (6-8 fates)
-Area is absolutely loaded with fates. My personal favorite place for levels 20ish-30. I leave in later 20's to 30's and head to Costa Del Sol. -South Shroud Below in 25-30 range also pertains to these comments.

LEVEL 25-30
EASTERN THANALAN: Highbridge (5 fates)
-"Attack on Highbridge Act1-4" lv.26: There is a multiple chain FATE here. You fight Quqirn and can rack up lot's of exp.
SOUTHERN THANALAN: Little Ala Mhigo (6 fates)
NORTH SHROUD: West of Fallgourd Float (4 fates)
*SOUTH SHROUD*: Quarrymill (3-5 fates)
-"The Courel King" lv.29: East of Quarrymill is an area called Taker's Rot, There is a 3 part FATE. Good exp!

LEVEL 30-35
SOUTHERN THANALAN: Area around Forgotten Springs (5 fates)
*EASTERN LA NOSCEA*: Costa Del Sol (7-8 fates)
-The Bloodshore is packed with great fates and frequent parties. Great place for this level range possibly even a little sooner.
-"It's Not Lupus" lv.32: Fight Cancer a giant crab. Lot's exp is gained from this fight and upon completion you receive a smallshell crab minion.
-The FATE with the bombs can get pretty annoying to target, you can activate a balloon to attract all the bombs towards you.
-The FATE with the Giant Aurochs can be hard as well because they die so fast, just try you best.
-Lv. 34-35 head over to Dragonhead in Coerthas.
CENTRAL SHROUD: Northeast of Bent Branch (4-5 fates)
SOUTH SHROUD: Camp Tranquil (4-5 fates)
COERTHAS CENTRAL HIGHLANDS: First Dicastrial Observatorium of Aetherial and Astrological Phenomena

LEVEL 35-40
*EASTERN LA NOSCEA*: South of Wineport
-Decent alternative to Dragonhead. (see below)
UPPER LA NOSCEA: Camp Bronze Lake
SOUTH SHROUD: Northeast of Camp Tranquil
-Come here around lv.34-35 once you are done with Costa Del Sol.
-Lots of great FATEs and lots of parties.
-There is a three part FATE here where you slay the dragon Svara for some great exp.
-Unfortunately PS3 users and low end computer users may experience crippling lag here due to large amounts of people. The snow weather is fairly constant and can make the game really crawl.
-The other spots on the list are still viable options.

LEVEL 40-50
MOR DHONA: Revenant's Toll
*NORTHERN THANALAN*: Cerulean Processing Plant
-FATEs of note are Dark Devices, and Cores. These are amazing XP.
WESTERN LA NOSCEA: Northwest of Aleport
OUTER LA NOSCEA: Camp Overlook
EAST SHROUD: Northeastern part of map
NORTH SHROUD: East and North of Fallgourd Float
SOUTH SHROUD: Small section Southeast of Quarrymill (Odin area)

Detailed map for locations

Credit goes to Eben Claire