Olivine City is a major port city situated on the west coast. Local attractions include the Olivine Gym and the Glitter Lighthouse, neither of which is operating normally. In addition, the S.S. Aqua, a ferry that sails to Kanto's Vermilion City and back, is not currently operational either. Silver meets you near the Gym, and mentions that the Gym Leader is tending to a sick Pokémon in the lighthouse.
Stop by the house south of the Gym. The Fishing Guru inside will give you the Good Rod, a mid-level rod that is capable of hooking more variety of fish than the Old Rod.
Time for a Trade
North of the lighthouse are two houses. The man in the western house would like to trade, and is willing to offer his Voltorb for a Krabby. Krabby are fairly common in these waters, especially with your new Good Rod.
Pay a visit to the Olivine Café to the west of the Pokémon Center. Speak to the Sailor inside to receive HM04 (Strength), a powerful move that lets a Pokémon move large boulders with ease.
The Lighthouse is easy to navigate. First, climb all the way to the fifth floor to gather the items there. Climb down to the fourth floor and drop through the hole near Lass Connie. Sailor Terrell attacks when you land on the third floor; defeat him and claim the Ether. Climb two flights of stairs to reach the inside area of the fifth floor, and battle Sailor Ernest. Grab the Great Ball lying in the corner and take the stairs to the sixth floor.
Jasmine, the local Gym Leader, is here tending to her sick Ampharos, Amphy. It would normally use its tail to illuminate the harbor, but has gotten sick and has not yet recovered to its full strength. Jasmine talks of a pharmacy across the ocean in Cianwood City that specializes in a form of non-traditional medicine. She cannot leave Amphy unattended, and requests that you retrieve the medicine for her. Teach a Pokémon how to Surf, then travel west from the city to Route 40.
In the time it took you obtain the Silver Wing, Lugia has appeared in the deepest cavern. Stock up on Ultra Balls, and bring a party that is capable of inflicting status ailments in order to wear it down faster. There are many items to be found throughout the caves, but if you are only looking for Lugia, start with the northeast island.
Step inside the cave, and take the stairs to the floor below.
Walk south to find a Carbos by the west wall, then climb down the ladder south of here. Grab the Max Elixer and Full Restore and backtrack to the island's entrance. Head south off of the rock ledge to find another Full Restore nearby. Turn east and trek across the northern rock ledge to get a Nugget. Jump the double-ledge to the east and climb the ladder for an Ultra Ball. Head back down and go south, jumping two single-ledges on the way. The boulder can't be moved from this side, so grab the Escape Rope lying here and climb the ladder to the southwest. This brings you to the southwest island's entrance.
Lugia in the deepest chamber
Walk east, jump the ledge, and take the northern path to reach the next part of the main cavern.
Follow the rock platform southward, and climb down the ladder to the waterfall cave. Pick up the Max Revive and surf to the south. Ride the waterfall to the bottom and enter the cave to the west. Save your progress; there's only one Lugia, so don't risk losing it.
Taking the ladder south of Lugia's chamber will lead you to the southwest island's entrance, but if you're done here, using an Escape Rope is easier.
Walk north along the trail and take the stairs to the main cavern.
Head north along the rock ledge to find a Calcium. Turn around and go south off of the ledge and loop around to the west. Moving the boulder here will take you directly to the southwest island's entrance. But instead, climb the ladder to the south.
Pick up the Guard Spec., then surf west to reach the exit to the southwest island.
The only items accessible from the southwest island are the Escape Rope and the Calcium, both of which would have been collected already.