How to Obtain Xerneas, Yveltal, and Zygarde in Pokémon Legends Z-A
Introduction
Xerneas, 
Yveltal, and 
Zygarde are three Legendary Pokémon introduced in Pokémon X & Y. Xerneas is known for representing life and rebirth, Yveltal is known for representing destruction and death, and Zygarde is known for being the "guardian of the ecosystem".
How to Obtain Xerneas
After winning the first rival trainer in the Infinite Z-A Royale and going back to the Hotel Z, you will be met by L, also known as Lysandre. He then talks about " The One That Gives" (Xerneas) and "The One That Takes" (Yveltal). To obtain Xerneas, you must prioritize the "The One That Gives" mission.
Firstly, you need to talk to Mable at the Pokémon Research Center. She will tell you that anomalous Alpha Pokémon have appeared in Wild Zones 3, 5, and 9, and that you need to catch or defeat the alphas.
Steelix
Gallade
PangoroAfter catching or defeating the three Alpha Pokémon, Mable then tells you that Xerneas is on track to appear in Wild Zone 11 and tasks you with catching it.
Once you have headed to Wild Zone 11, you will see a glowing marker beside the river. By interacting with it, a Level 75 Xerneas will appear in front of you.
Then, when Xerneas appears, you will need to catch it. Xerneas is a Fairy-type Pokémon and is weak to Poison and Steel-type moves. If you accidentally knock it out, then you will have to redo the encounter and try to catch it again. After catching Xerneas, the "The One That Gives" mission will be complete.
How to Obtain Yveltal
After winning the first rival trainer in the Infinite Z-A Royale and going back to the Hotel Z, you will be met by L, also known as Lysandre. He then talks about " The One That Gives" (Xerneas) and "The One That Takes" (Yveltal). To obtain Yveltal, you must prioritize the "The One That Takes" mission.
Firstly, you need to talk to Grisham and Griselle. They will tell you that it would be wise to reach out to Quasartico and to meet them at Quasartico Inc. Building.
After heading to the Quasartico Inc. Building and talking to Vinnie, you will be tasked to fight Rogue Mega 
Victreebel, 
Hawlucha, and 
Tyranitar again in a simulation made by a tool in the lower levels of the building.
When you have defeated the three Rogue Mega Pokémon, Vinnie then tells you that Yveltal is on track to appear on the rooftop of Galerie de la Lune and tasks you with catching it.
After heading to the rooftop of Galerie de la Lune, you will see a glowing marker on top of the dome. By interacting with it, a Level 75 Yveltal will appear in front of you.
Once Yveltal appears, you will need to catch it. Yveltal is a Dark/Flying-type Pokémon and is weak to Electric, Ice, Rock, and Fairy-type moves. If you accidentally knock it out, then you will have to redo the encounter and try to catch it again. After catching Yveltal, the "The One That Takes" mission will be complete.
How to Obtain Zygarde
After catching both Xerneas and Yveltal and heading back to Hotel Z, you will talk to L, also known as Lysandre, in front of the entrance and get the "To Keep The World in Balance" mission.
Firstly, you will need to fight L and his Pokémon in a trainer battle.
L uses this team of Pokémon:
Pyroar (level 78) - Normal/Fire
Florges (level 78) - Fairy
Sableye (level 78) - Dark/Ghost
Garbodor (level 78) - Poison
Noivern (level 78) - Flying/Dragon- Mega
Gyarados (level 80) - Water/Dark
Once you have beaten L, you will be approached by Zygarde, and L will task you with meeting it in the middle of Wild Zone 20. Zygarde will be standing under the Prism Tower, and by interacting with it, you will start a battle with Zygarde.
Zygarde is a Dragon/Ground-type Pokémon and is 2x weak to Dragon and Fairy moves, and 4x weak to Ice moves. Zygarde will be Level 84 and have three phases in this battle.
Phase one: 10% Zygarde
In the first phase, Zygarde will be at 10% power and mainly using its Thousand Arrows move.
Phase two: 50% Zygarde
After beating Zygarde at 10% power, it will regain its HP and go into its 50% power form. It will often use the moves Rock Slide, Crunch, Dragon Pulse, and its ultimate attack, Land's Wrath.
Phase three: 100% Zygarde
Once you have defeated Zygarde at 50% power, it will regain its HP for the last time and transform into its 100% power form. In this final phase, Zygarde will be mainly using Thousand Arrows and Thousand Waves. It will also alternate between two ultimate attacks: Core Enforcer and Land's Wrath.
Finally, after beating the three phases, you will be able to catch Zygarde in any Pokéball you would like, as it has a guaranteed catch rate
When you have caught Zygarde, you will be approached by L, and he will give you the Zygarde Cube item, which allows you to change Zygarde's form between the 10% and 50% form. Zygarde changes into its 100% form when it goes under 50% HP in a battle.
Once you have finished talking to L, the "To Keep The World in Balance" mission will be completed.