The Ghost-Type has advantages over Psychic types, although Gastly, Haunter and Gengar are Poison as well so it is leveled outwhn using those three. They also have a huge advantage over fighting types since they basically can't be hit by them. The same goes for Normal type Pokemon, but then the Ghost Pokemon better have something else then only Ghost type moves or it will be an eternal battle :P. Ghost attacks are also super effective against other Ghost Pokemon.
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Fighting Pokemon are weak against Psychic types and Flying types. Ghost Pokemon are also a good choice against Fighting types since Fighting and Normal type attacks don't effect them.
Psycic Pokemon are weak against Ghost type, Bug type and Dark type. Psychic types are good against Fighting and Poison types since Pokemon Gold (and onward)
Since Lugia is a Psychic/Flying type pokemon, I would try Ghost types and electric types. Ice types are also a good choice
well, spiritomb is a ghost type and ghost types are weak against ghost and pychic. so, moves like bite, shadow ball and pychic might be a good place to start.
Gengar is a ghost/poison type so... ghost and dark types are super effective against ghost types. ground and psychic types are super effective against poison types