Help me, BLM-senpai!

[Seraphor]

BLM at 80 is as smooth and fluid as it has ever been, but you've got a little way to go before then.

At level 60 you get Fire IV, which is your main reason to keep Enochian up at all time.
Remember that Bliz3 and Fire3 are your transition spells that put you into UI/AF3, at which point you'll use Enochian, and never let it drop.

For example, if you start with Bliz3, then hit Enochian, you'll have time to get your Thunder out, and then Bliz4 to generate your Umbral Hearts (free fire spells) before you hit Fire3.
In AF with Enochian, it's Fire4 spam until you're about to lose Enochian or MP. If Enochian is less than 6s, use Fire1 to refresh it, if your MP isn't enough, hit Bliz3.
And repeat.
Swiftcast and Triplecast as necessary to facilitate movement. Firestarter proc can be used for movement or AF refresh depending on when it comes up. Thundercloud proc for movement, or whenever you get a chance to squeeze it in if you don't want to lose it.
Just remember, never lose Enochian.

That's the simple version at least, from there you can either learn yourself or look up more detailed guides.