Battle Moveset

A Good Moveset for Lugia would have to be this:

Attacks:

Substitute
Calm Mind
Recover
Psychic

Items Attached:

Leftovers

Preferred Nature:

Calm

Strategy Using Lugia

Lugia's special defense is already very high, high enough to survive Super effective special attacks. The EV's focus on building its defenses.
Assuming its EV's are maxed, HP should be 416, Defense will be 359 and Special defense will be 379. Though you could make Lugia into a special wall, with +special defense nature and 252 EV's into special defense to give it a 447 special defense.
Calm mind, however, helps boost special defense anyways, which after one use, makes its special defense 569 (379 x 1.5 rounded up).
Substitute allows for those 101 HP substitutes which can survive moves like Night shade and Seismic toss. The substitute breaker, bonemarang will not work either due to that move being a ground type and Lugia being flying. Though Rock blast will be a nice counter to its subbing and has a type advantage over Lugia. Leftovers is probably enough to nullify the need for Recover, but that exposes you to a Jolteon's Thunder wave. So Leftovers could always be the better choice instead of lum, since it frees up a move slot (get rid of recover). So to go along with its tanking, it can special sweep so its not completely exposed to dark types like the above moveset is (that's if you pick Psychic over Aeroblast).
Lugia could use the Moveset: Thunderbolt, Icebeam, Calm mind and Substitute, which allows for it to stick around for a long time against a team that has no hazers or pseudo hazers. Lugia can also T-wave, Whirlwind, Light screen and reflect are four possible moves that can be used on Lugia (not necessarily in the same moveset though). Since Lugia has a 4x resistance to brick break, if your opponent does attack with brick break and destroys Light screen/Reflect, it won't recieve much damage and will be able to set up another Reflect/Light screen easily.

EV Corner:

Max out the EVs in Defense and HP for Lugia with any remaining in Special Defense

Strategy Against Lugia

Lugia is potentially the hardest Pokémon to defeat, with all Tank stats above average and access to 101 Subs (meaning it's Substitutes can take two this from moves such as Seismic Toss and Night Shade in case you didn't guess) as well as access to Calm Mind to boost that monstrous Special Defence and Recover for the HP, the really is no easy way to defeat Lugia.
- Taunt is your friend, and one of the few friends that you have for taking down this beast. If you can sneak it in before it uses Substitute or any other utility/defensive move you may be in with a chance.
- Status's. Very necessary. Toxi-stalling (with Protect and Substitute working with leftovers) could be an effective strategy, but you may find that Lugia has a Lum Berry or Rest, which may cause major issues for you. Not to mention the fact that if Lugia switched out the effect of Toxic would be nullified. On the other hand, Paralysis could cut away that high speed and the possibility of an offensive, however it would still require a very high offensive stat and some super effective hits to take down this beast. Sleep Powder and Spore are the two best status moves available for taking down Lugia. Disable the beast, set up a Substitute (to cover you if it awakes) and beat down Lugia.
- Hazing may be necessary, since a Lugia using Calm Mind will not only be impenetrable by special attacks, but it'll also be a very deadly special attacker. The ability to get rid of its Calm Minds will be necessary if it starts to stat boost, otherwise you'll have lost the battle.
- Luck, and lots of it, is another way to go. If you have access to Critical hit techniques (and use the rarely used Focus Energy + Scope Lens combination) you may be lucky enough to KO Lugia as it switches in. If not, you'll need to be faster than Lugia in order to get a strong hit in. You could also go for the Sand Veil + Bright Powder strategy and evade offensives while picking away at its HP, but you'll need a Pokémon that isn't fragile to pull this off.
Make sure to assault the weaker (and it is by no means weak) Physical Defence rather than go for Special Defence. Super effective moves on Special Defence won't have a huge effect on Lugia's HP, while a strong Rock Slide or Shadow Ball will have an effect. To be honest, if you do wanna go on an offensive against Lugia, use a Legend of your own. It's only fair that if your opponent get to use one, you should get to use one as well. The mighty Mewtwo can hurt Lugia's Special Defence but is by no means unable to hurt Physical Defence as well. On the other hand, you could use Lugia's opposite, Ho-oh, although you may put yourself at risk to Lugia's water attacks. Groudon or Kyogre are effective choices since they have the weather on their side, however they are outmatched in speed. Rayquaza, if given the chance to use Dragon Dance, will also be an effective counter. Either way, if using a non-legendary Pokémon, pray for luck to be on your side and try to use moves such as Shadow Ball and Rock Slide.

Contest Moveset

A good Contest Moveset for Lugia would have to be this for the Cool Contest best with Lonely, Adament, Naughty or Brave Nature:

Attacks:

Rain Dance
Thunderbolt
Thunder Wave
Hyper Beam

Items Attached:

Red Scarf

Strategy Using Lugia

If you do the attacks in this following order you should have very few problems; 1st - Rain dance, 2nd - Thunderbolt, 3rd - Rain dance, 4th - Thunderwave, 5th - Hyperbeam

Locations in Games

Ruby/Sapphire/Emerald Trade from FR/LG/E (R/S), Navel Cavern (Emerald) Colosseum Trade from FR/LG/E Fire Red/Leaf Green Navel Cavern

Animé Appearences

Lugia has had a few Animé Appearences. First it was in the Orange Islands and arrived when Lawrence III brought together Articuno, Zapdos & Moltres. With Ash's help, it was able to qwell the fighting. After that, two were in the Whirl Islands and under attack by Butch & Cassidy. However with Ash & Richie's help, they were freed Movie 2: The Power Of One! Episode 222: The Mystery Is History! Episode 223: A Parent Trapped! Episode 224: A Promise is a Promise!