FFXIV is very streamlined since late HW with the way mechanics are telegraphed and named as well as how arenas are designed.
The most important thing for being able to learn mechanics is actively paying attention. Not just pushing buttons, seeing everyone run in a certain direction and blindly following them but really watching what's going on around you. That's how you learn patterns and get a good feeling for how mechanics work so that when doing any content blind you can already make an educated guess.

Boss raises the left arm and it glows? Problably means a huge side cleave.
Boss cast "double" something? Probably a mechanic that hits twice, dodging into the first safe zone works almost always.
Boss glows in one color while things around it glow in the same and a different color? Probably means standing on the differently colored thing.
Boss tethered you to something/ someone? Running to Narnia works 95% of the time to break the tether and avoid something nasty.

Do content, pay attention, watch your surroundings, keep a close eye on the boss' castbar and watch other players and you'll soon learn the mechanics. Worst case you die, get ressed and can try again.
And thanks to how streamlined most are, you'll have an easy time even when in a dungeon you don't remember doing but recognize the usual tells. It doesn't matter if it's boss A or D doing a side cleave, a side cleave is a side cleave.