The first view of the Aether Foundation is in the opening cutscene where it shows Lillie running from various Aether Foundation employees after taking cosmog. It ends with Cosmog creating a portal for it and Lillie to escape.

Location: Diglett's Tunnel

Your next active encounter with the Aether Foundation is in Diglett's Tunnel. When travelling through there, you will encounter two employees who will greet you. They will inform you that Team Skull had been riling up Pokémon in the area and that Team Skull often interfere with Pokémon.

Location: Memorial Hill

When you get to Memorial Hill later in the game, you will encounter an Aether Foundation employee and their boss arguing with Team Skull over a Slowpoke. The Aether admin requests that you fight in his stead to stop Team Skull from taking the tail from the Slowpoke. If you do, they suggest going the Hano Grand Resort hotel so you can talk to him.

Location: Hano Grand Resort

When you reach the Hano Grand Resort, you will meet the Aether admin you helped in Memorial Hill. He will introduce himself as Faba and will offer an invite for you to go visit the Aether Paradise, a special island created by the Aether Foundation in the middle of the Alola Region. You, along with Hau, then go to the boat to go to the Aether Paradise!

Location: Aether Paradise

When you get on the boat to the Aether Paradise, Faba will show you around. He will note that there is a jamming signal in the Aether Paradise so that nobody can use Poké Balls and capture the Pokémon in the area. They want to create a safe haven for Pokémon where people such as Team Skull cannot get them. Faba then introduces you to Wicke, another of the Branch Admins, who takes you and Hau up to the special Conservation Area.

In the Conservation Area, you then meet the character Lusamine who explains how she is creating the Aether Paradise to look after all Pokémon. However, while talking, an Ultra Wormhole appears and an Ultra Beast comes out of it, leaving you to have to battle it. It gets a Defense boost

Ultra Beast

???

Level 27

Attacks: Clear Smog Psywave Headbutt Venoshock

Hold Item: No Item

When defeated, Lusamine will explain that she needs a Pokémon to help find these Ultra Beasts as it was stolen from her. She then instructs Wicke to send you to Ula'Ula Island

Location: Aether Paradise

The next encounter with the Aether Paradise comes much later in the game after Lillie has been kidnapped by Team Skull. Gladion makes the connection and states that Lillie has been taken to the Aether Paradise and that you, he, and Hau must go there to rescue Lillie and stop their plan.

When there, the Aether Foundation employees will now challenge you to battles as they have been given orders to stop anyone from going through. As you progress, you will learn that you can't go to the basement so you have to go up and try to get the key from Branch Chief Faba. Eventually, you go to the Secret Labs in the basement to get the information about Cosmog being an Ultra Beast, with the help of Wicke.

Aether Branch Chief Faba	Slowbro	Bruxish	Hypno
Battle Type Items	Level 39	Level 39	Level 40
	Attacks: Yawn Psychic Amnesia Headbutt	Attacks: Aqua Jet Crunch Psychic Fangs	Attacks: Psychic Hypnosis Disable Nightmare
	Hold Item: No Item	Hold Item: No Item	Hold Item: No Item

Location: Aether Paradise

As you battle through, you discover that Aether Foundation and Team Skull have in fact been working together, with Guzma there in order to help Lusamine achieve her goal. Once Guzma has been defeated, you, Gladion, and Hau are able to get into the main room where Lusamine is, with Lillie and Cosmog. Lusamine intends to use Cosmog's energy to open an Ultra Wormhole to bring in the Ultra Beast, not caring that it's going to kill Cosmog and chastising her children, Lille and Gladion, for stealing the Pokémon Cosmog and Type: Null from her. Lillie tries to convince Lusamine to stop but Lusamine succeeds in using Cosmog's powers to open the wormhole when an Ultra Beast comes out of it. Wormholes begin to open all across the Alola Region.

Following this, Lusamine then challenges you to a battle.

Aether President Lusamine	Clefable	Lilligant	Mismagius	Milotic	Bewear
Battle Type Items	Level 41	Level 41	Level 41	Level 41	Level 41
	Attacks: Moonblast Cosmic Power Metronome	Attacks: Petal Dance Teeter Dance Stun Spore	Attacks: Shadow Ball Mystical Fire Power Gem	Attacks: Hydro Pump Safeguard Flail	Attacks: Baby-Doll Eyes Hammer Arm Take Down
	Hold Item: No Item	Hold Item: No Item	Hold Item: No Item	Hold Item: No Item	Hold Item: No Item

After defeating Lusamine, the wormhole will start to close and the Ultra Beast disappears into it, with Lusamine and Guzma both following suit. After the wormhole is closed, it seems Cosmog has evolved into Cosmoem. Lillie is eager to find her mother and stop her from doing any more damage but Galdion & Wicke mention that perhaps the power of Solgaleo/Lunala can awaken Cosmoem and create a wormhole to get Lusamine back. Lillie changes her look and Gladion wishes you well on your journey by giving you the Moon Flute/Sun Flute.

Location: Ultra Space

Once you have got the other flute from Exeggutor Island and made it to the Altar of the Sunne/Altar of the Moone, you awaken the powers of Solgaleo/Lunala and have Cosmoem evolve into it. It then creates an Ultra Wormhole that lets you go to the mysterious Ultra Space. There, you meet Guzma who explains how he tried to catch UB01, but it instead took control of its mind and now he knows fear, and that Lusamine is too far gone. Further into the space, among all the Nihilego, you will find Lusamine. Lusamine is very happy in the world of Ultra Beasts where the only thing that exists is the love between her and Nihilego. She then goes on a rant about how if things aren't beautiful, regardless of if they're the daughter or a rare Pokémon, then she just doesn't care. Lillie then chastises her mother's way of thinking, but Lusamine counters with her thoughts being like a trainer's. She releases Nihilego from a Beast Ball, but then mysteriously merges with it. Then, the powerful Lusamine/Nihilego hybrid challenges you to a battle. All of Lusamine's Pokémon have got stat boosts in various stats. Clefable in Sp. Def, Mismagius in Speed, Lilligant in Special Attack, Milotic in Attack, Bewear in Defense

Aether President Lusamine	Clefable	Lilligant	Mismagius	Milotic	Bewear
Battle Type Items	Level 50	Level 50	Level 50	Level 50	Level 50
	Attacks: Moonblast Cosmic Power Metronome Moonlight	Attacks: Petal Dance Teeter Dance Stun Spore Leech Seed	Attacks: Shadow Ball Mystical Fire Power Gem Pain Split	Attacks: Hydro Pump Safeguard Flail Recover	Attacks: Baby-Doll Eyes Hammer Arm Take Down Pain Split
	Hold Item: No Item	Hold Item: No Item	Hold Item: No Item	Hold Item: No Item	Hold Item: No Item

After defeating her, Nihilego will separate from Lusamine. Lusamine then passes out, stating how beautiful Lillie is, but becomes very weak. Later on, Lillie takes Lusamine to Kanto to meet Bill, who also once combined with a Pokémon, in order to try and help her.

Location: Aether Foundation

After defeating the Aether Foundation, the employees are all apologetic and can still be found on the paradise trying to care for Pokémon since they didn't hold Lusamine's ideals. Wicke helps with the aid in hunting down the Ultra Beasts, Gladion gives you a Type: Null and Faba will battle you again

Aether Foundation Faba	Slowbro	Bruxish	Raichu	Alakazam	Hypno
Battle Type Items	Level 61	Level 61	Level 61	Level 61	Level 61
	Attacks: Yawn Psychic Amnesia Ice Beam	Attacks: Aqua Jet Crunch Psychic Fangs Screech	Attacks: Psychic Thunderbolt Quick Attack Nuzzle	Attacks: Psychic Psyshock Disable Dazzling Gleam	Attacks: Psychic Hypnosis Focus Blast Dream Eater
	Hold Item: No Item	Hold Item: No Item	Hold Item: No Item	Hold Item: No Item	Hold Item: No Item