Stick to low level/one-boss FATEs that way you can knock them out <1min, FATEs adjacent to the cities are perfect for this
Avoid high level/mob FATEs, Outer La Noscea is the worst for this
Gauge what FATEs are worth travelling to vs having it expire or downed by someone else, Southern Thanalan is the worst for this
Make sure the right weapon is equipped and have your choco out throughout.

Use Party Finder in a zone, until yu have got (a few) atmas

It gets slightly easier once yu are doing your first Atma book because it will require yu to revisit these zones and yu can down a FATE on the way to your book kill, make sure your zenith wep is equipped, not atma wep