Update: As of 5.3 this is obsolete. You can only buy Augmented gear now. See this post for the latest info.
I made this post so that I could easily upgrade my Ironworks and Shire Gear. There are a few places to find the info, but they are not convenient to refer to. This is designed to be easy to use.
Note: Next time I visit Rhalgr’s Reach, I’ll add the guide to upgrade the level 70 Scaevan gear. The purchase and upgrade NPCs are all at the same counter for the Scaevan Gear.
Upgrade Ironworks – Level 50 Gear
Purchase of Ironworks gear is in Revenant’s Toll, Mor Dhona. Purchase the gear from Auriana at the counter on the upper level. Auriana also has Upgrade Mats under the Other category of her store.
Purchase a single Carbontwine for Armor and Weapons, and a single Carboncoat for Accessories.
Travel for the Ironworks Upgrade
Then take your gear piece, unequipped, and your mats all the way to the North Shroud. The closest Aetherite is Fallgourd. Once there, travel North East along the road to Hyrstmill, at the far NE of the map and speak to Drake who can upgrade your gear. What a pain. Thankfully, later gear involves far less travel for the upgrade.
Upgrade Shire – Level 60 Gear
Purchase and upgrading of Shire Gear is done in Idyllshire, but you probably already know this. Here’s where you purchase the gear and the upgrade materials.

The upgrade NPC is at the stall across the street. To upgrade Weapons use Illuminati Deepest Gobdip.

To upgrade Armor use Illuminati Tautest Gobtwine. One spool for each item of course.

To upgrade Accessories use Illuminati Darkest Gobcoat.

Scaevan Gear – Level 70 ilvl 390
Here’s a quick summary of how to purchase and upgrade Scaevan Gear. This gear is now purchased using Tomestones of Poetics, since Shadowbringer’s has released. Tomestones of Genesis have been removed from the game.
Another Interesting Thing
The Large Beaver Minion has the wrong kind of teeth. I’m not clear why the FFXIV designers didn’t look up Beavers and see that they are essentially Rodents with Incisors rather than Canine teeth. But here they are:
I’m not saying he’s not cute, but his teeth are wrong.
Oh well. May you walk in the light of the Crystal.
:ww