I have a bunch of quests that require me to do duty finder things that say you need an average item level of like 60 or 70 or something and I'm only level 51, is there a way to equip higher-level items than your player level?

@Felthry You can! Made a picture to explain how to tell easier.

You can buy a lot of the gears with tomestones of poetics (Mor Dhona for lvl 50, etc.)

You can buy with gil too, vendors in the hub cities or zones close to your level will sell gear that you can use to fill in gaps.

@Eon question relating to those tomestones, by the way: what are tomestones of mendacity and genesis that are showing up in my currency window and claiming to be "no longer valid"? (I don't have any of them, I'm just curious)

@Eon also what's this about "reset in 8h 47m"? do you lose tomestones every day or something?

@Felthry The no longer valid message is because those types are older and getting rotated out.
Every time there's a patch numbered like 4.0,4.1,4.2,etc. they add a new type of tome (to go with the new set of gear added in that patch) and rotate out the old ones.

Poetics are the general catch all for content below the current level cap (80 now), so items for level 50,60,70 characters are bought with poetics and the 80 stuff is the new tomes that get added.