This is just a quick note to address that all threats and all synergies are written for the late game unless stated otherwise.
Don't expect to win early games as this is a hyper-scaling build and is very bad for the first 3-5 games.

This list will include only the augments you should take for most value + a few augments that are good in case the better augments don't show up.
The augments will show up from best to not as good, best being at the top.

BEST PRISMATIC AUGMENTS:

[*] Spirit Link: 40% of damage dealt to your ally is redirected to you, and 40% of healing they recieve is given to you as well.

[*] Goliath: Become large, gaining 35% Health and 15% Adaptive Force.

[*] Windspeaker's Blessing: Your Healing and Shielding also increases the target's Armor and Magic Resist for 3 seconds.

[*] Circle of Death: Healing and Regen you do deals a portion of the value in Magic Damage to the nearest enemy champion.

[*] Mystic Punch: Your Attacks reduce your cooldowns by 1 second.

[*] Mad Scientist: On Round start you grow large (Adaptive Force and Health) or tiny (Ability Haste and Move Speed).

[*] Back to Basics: Gain increased damage, Damage, Healing, Shielding, and Ability Haste, but you cannot use your Ultimate Ability.

[*] Ultimate Revolution: Once per round, refresh your Ultimate Ability after casting it.

[*] Wisdom of Ages: Gain 1 level now, and 1 bonus level every other round. Your level cap is removed.

BEST GOLDEN AUGMENTS:

[*] Dawnbringer's Resolve: Upon dropping below 50% Health, heal for 30% max Health over 3 seconds.

[*] Willing Sacrifice: When your ally drops below 30% Health, trade some of your health for a Shield on your ally.

[*] Celestial Body: Gain 1000 Health, but you deal 10% less damage.

[*] Banner of Command: Gain the Banner of Command Summoner Spell.
Banner of Command empowers your ally for a brief duration.

[*] Restless Restoration: You constantly heal while moving, increased by distance travelled.

[*] Die Another Day: Gain the Die Another Day Summoner Spell
Die Another Day creates a zone where no unit can die for 4 seconds.

[*] Rabbel Rousing: Using an Ability heals you for 2% max Health.

[*] Recursion: Gain 60 Ability Haste.

[*] Perseverance: Gain hugely increased Health Regen, which is further increased while low on Health.

[*] Shrink Ray: Your Attacks reduce an enemy's damage by 15% for 3 seconds.

[*] Vengeance: Gain massively increased damage and Omnivamp while your partner is dead.

BEST SILVER AUGMENTS:

[*] First-Aid Kit: Gain 20% Heal and Shield Power.

[*] Virtuous Cycle: Your Heals grant extra Shield and your Shields grant extra Healing.

[*] Sonic Boom: Buffing, Healing, or Shielding your ally deals damage and Slows enemies surrounding them.

[*] Juice Box: Each round, you and your teammate get an additional Juice for free.

[*] Contract Killer: Each round, mark an opponent to take 15% more damage and grant an extra 350 gold on death.

[*] Evocation: Gain the Evocation Summoner Spell
Evocation lets you channel to restore Health and Mana.

[*] Mind to Matter: Increase max Health by half of your Mana.

[*] Ocean Soul: Gain the Ocean Soul, granting high Health and Mana regen after damaging enemies.

[*] Guilty Pleasure: Immobilizing enemy champions restores 4% max Health.

I will discuss a few tips and tricks that will hopefully give you an extra little advantage in your games.

[*] Tahm Kench's Q can be buffered through cc allowing him to be able to deal damage/heal while being cc'd (for example, you can buffer Q during Warwick's ult to guarantee a hit. Same thing for Vi's ult and many more).
Using this concept, you can guarantee a Q hit on an enemy champion you swallowed by using Q just before the 3 second ult timer ends (make sure you don't recast your ult otherwise you won't be able to cast your Q).

[*] While it is written in his Q (Tongue Lash) tooltip, many people either forget, ignore, or just don't know that Kench's tongue's length can be increased with champion size.
This means, if you take the Goliath augment, you can launch your tongue about three times as far as you usually can.

[*] Even though it is also written in his Q (Tongue Lash) tooltip, not many people know that when an enemy champion has three passive (An Acquired Taste) stacks you can press R (Devour) while your Q is in mid air to swallow said enemy champion from afar.
This can be even more useful in a scenario where you have Goliath and enemy team very strong as a duo but weaker alone, allowing you to separate them without risking death.

[*] I am not sure if Riot meant for this to happen or if it's a bug but, when you ult your teammate before the round starts, there is a wierd interaction with Moonstone allowing you to recieve a permanent shield equaling about the same as your actual ult shield. Abuse this for about an extra 1,000 health before each round starts.