Disclaimer: This is an accumulation of thoughts, including my own, from several people throughout our clan over several seasons of play. Each player may have their own ideas or suggestions on how to go about augmenting items, and there really isn't a wrong or right way of doing it. That's the beauty of the game!
To hopefully shed some light on new or inexperienced players on the process of augmenting, and how to best optimize getting their gear augmented.
Unlike Ancient items and gear in general which mostly rely on RNG, augmenting your gear is a way to boost your stats independent of luck and paragon. Augmenting can only be done on Ancient (or Primal) Legendary pieces. With time and skill, you can level a legendary gem to consume in the cube to directly place a stat of your choice on an Ancient (or Primal) Legendary of your choice.
How to Augment:
You level a gem of your choice to whatever level you'd like. You take that gem, as well as 3 Flawless gems (Topaz for Intellect, Emerald for Dexterity, Amethyst for Vitality, Ruby for Strength) PLUS the Ancient/Primal of your choice PLUS that legendary gem you leveled. When the augment is completed you will get 5 of that stat per level of that gem. So for a level 60 gem, you will get 300 stat points. Please Note: It does not matter which of the legendary gems you choose to level and use, as the special effect on the gem will not transfer to the gear. Typically you stay away from leveling Gem of Ease, Boon of the Hoarder, Gogok of Swiftness, Iceblink, and Invigorating Gemstone for augmenting as those gems have a level cap.
There are minimum level requirements per item, seen here (also found in-game in the cube recipes):
That is the *general* info on Augmenting.
Where to start in terms of efficiency?
Most recommend starting at not less than a level 60 gem. Ideally you will want to get a few (2-3 or more) gems to whatever level in a quick manner (quick being greater rifts being completed in the less than 2-4 minute range). Then, grouping up to get the gems to that point if you cannot do so solo (or can, but not in a quick manner). There currently isn't a shortage of people willing to do speed runs to help others get gem levels, as well as event nights where the sole objective is to help with gem leveling.
What items to pick first?
Well, you certainly don't want to augment just *any* Ancient, unless you have very little choice. You will want to select one that is close to perfect, or rolls almost all of the desired affixes on the item. This will cut down on having to re-level a gem in the event of an upgrade (and the more poorly rolled an ancient, the higher probability of that happening.) I personally prefer to augment items that have an easier time rolling this way: chest, legs, boots, bracers. Hopefully by the time I have finished augmenting these items, I will have an ancient for the remaining slots worthy of getting an augment. If I do not, then I typically put a 55-60 on them, level another gem as high as I can, and wait to use it until I do. This ensures I still get the max power creep possible while still hunting for an upgrade.
2.5 Update: Go ahead and augment any Primals you get since they roll 100% of the stat varieties they have - especially if they roll the majority of the affixes you desire for your build.
Note: You may override an existing augment on an Ancient, but the gem used must be a higher level than that of the previous gem used on that item. I.E. I have augmented my gloves with a 55, I must use a level 56 or higher gem to replace that augment.
If you have any questions or feedback, please feel free to discuss below!
Any frequent questions will likely be added to the guide.