Instructions
Go to Goldenrod City and defeat Team Rocket and its leader at the Radio Tower. The manager of the Radio Tower will give you the Silver Wing as a gift for saving him.
Go to Ecruteak City and fight the five Kimono Girls at the Dance Hall. Be ready, because you will not have a chance to change Pokemon or heal between battles. They will give you the Tidal Bell after you defeat them and then tell you to meet them at the Whirl Islands.
Go to Olivine City and head south on Route 41 until you reach the Whirl Islands. You will need the Whirlpool ability to get past the whirlpool blocking the route.
Land on the northwestern island, and make your way through the maze until you reach the lowest level. You will see a monk, who will only let you pass if you have the Silver Wing. Once you pass the monk, the Kimono Girls will greet you if you have the Tidal Bell. They will perform a dance to summon Lugia, and he will appear.
Engage Lugia in battle and wear him down as much as possible without causing him to faint. Put him to sleep or freeze him, and start throwing balls until you capture him. Dusk Balls, Heavy Balls and Ultra Balls will have a greater chance of success, but if you run out of these balls, use any others you may have.