Lugia

As a cover legendary, Lugia is somewhat disappointing like Ho-Oh. Its stats are heavily slanted towards defense, which is not ideal in raids. Because it is a legendary species, Lugia can’t use its bulk as a gym defender, so its role is primarily as a bulky Psychic/Flying-type attacker. Unfortunately, Lugia often finds itself outclassed despite Aeroblast's incredible damage, as many other Psychic or Flying attackers have significantly higher Attack. Its lack of Flying-type fast move further limits its potential.

Apex Purified Lugia wields the obscenely powerful Aeroblast++, the strongest charge move in the game. This somewhat makes up for Lugia's low Attack stat and raises its DPS to respectable levels. However, its Shadow form is the real deal, and it is not recommended to purify Apex Shadow Lugia if you care about its raid performance.

Best PvE Offensive Moveset

• Extrasensory is Lugia’s best quick move, targeting Poison and Fighting-types for STAB effective damage. Dragon Tail isn't really worthwhile for Lugia.

• If you're going for a Flying Type Charged Move, which you should, then you've got two main choices. Fly out-paces Aeroblast and Sky Attack, though Aeroblast++ is slightly stronger than Fly. That said, don't purify an Apex Shadow Lugia unless you have a really good reason, please!
• Future Sight can work if you're determined to stick with a full Psychic Type moveset, though there are genuinely better options for that role.
• Hydro Pump has little to no use for Lugia.

With the Season 8 buff to Dragon Tail, its incredible damage output now makes it the superior option to Extrasensory despite having slightly less energy gain. Dragon Tail is especially preferred in Master League due to the ubiquitous use of Dragon-type Pokemon. 

Fly is a very efficient move, benefits from STAB, and costs the least amount of energy, and vastly outclasses Sky Attack. Aeroblast, Lugia's signature move, possesses immense power, blowing apart enemies even when resisted; it has a steep 75 energy cost but Lugia's bulk makes it very realistically reachable. Hydro Pump can be used to cover threats like Rhyperior, but otherwise a resisted Aeroblast deals just about as much as a neutral Hydro Pump. Future Sight and Return lack both the power and coverage to be worthwhile.

Aeroblast++, the signature move of Apex Purified Lugia, is functionally identical to Aeroblast in PvP.