Instructions
Speak to the Pokemon in the town square then head over to the pond and speak to Wishcash to receive "HM Dive." This also opens up the "Stormy Sea" Dungeon which is one of the requirements for getting Lugia.
Complete the Stormy Sea dungeon then travel to Northern Range and defeat Latios. Doing so unlocks Pitfall Valley, where you must rescue Latias. Once you have done this, both Latios and Latias join your team, which is another requirement for getting Lugia.
Befriend the three legendary birds "Articuno, "Zapdos" and "Moltres." This can only be done after the main story line has been completed and is the third requirement for getting Lugia.
Speak to the Pokemon in the town square named Alakazam. He informs you about strange happenings near the legendary birds friend area. Put a water Pokemon in your party and go speak to the legendary birds. You will receive a symbol that allows you to enter a new dungeon called the "Silver Trench."
Battle through all ninety nine floors of Silver Trench and you will reach Lugia at the end. Defeat it in battle and it will join your party.