What gear should I get next?

So I have all Augmented Shire gear and weapon. I am level 64 and not sure what the next highest gear is.

The augmented shire gear will hold you out until level 65, when you can earn gear from Bardem's Mettle. That is the first gearset that, stat-wise, will be better than your current gear.

You won't see any upgrades on Augmented Shire until the level 65 dungeon, Bardam's Mettle. For non-tanks, that tends to be minor upgrades over Augmented Shire and I wouldn't worry about it a whole lot.

The level 67 dungeon drops will be upgrades on every slot, and you can get them fairly easy by running it multiple times. One non-weapon piece is guaranteed and will be automatically placed in your inventory after the run, plus there are a lot of drops so it's relatively easy to get. There is another upgrade set in the level 69 dungeon.

You'll get a set of gear including a weapon for doing your level 70 class quest, so do that. That's your 70 starter gear, to be replaced by gear brought with Tomestones of Verity after you complete the main story quest "Stormblood".