Here's how you actually catch Lugia:
1. Beat the Elite Four and become the Champion.
2. Go to the Whirl Islands. This location is accessible by boat from Olivine City.
3. Talk to the old man in the lighthouse. He'll tell you about Lugia's location.
4. Find the underwater cave. You can enter it by diving off the nearby cliffside.
5. Go inside the cave. There will be a few trainers to battle inside.
6. Lugia will appear at the end of the cave. You can then battle and catch it.
Important Tips:
* Prepare strong Pokémon. Lugia is very powerful, so make sure you have a team that can handle it.
* Bring plenty of Ultra Balls. Lugia is a Legendary Pokémon, so it will be difficult to catch.
* Use status moves. Moves like Thunder Wave or Freeze-Dry can help weaken Lugia and make it easier to capture.
I hope this helps! Good luck catching Lugia.