Rikku, Kimahri, Yuna and Wakka's best weapons come with DOUBLE AP abilities once fully powered. You'll need to perform different tasks as well as obtain the corresponding items (crest, sigil and the Celestial Mirror) in order to recieve them, though. When you gain someone's legendary weapon at first, it bears only one ability: No AP. Getting these weapons powered up will not only boost your team's power accordingly, but also allow your Aeon's to do 5 digits worth of damage and concieve 5-figure HP. [I've put the names of the Aeons that will be affected]
And yes, you will be needing the Airship.
EVERYONE'S CELESTIAL WEAPON COMES WITH ABILITIES BREAK DAMAGE LIMIT AND TRIPLE OVERDRIVE.
Yuna: Nirvana - Double AP, One MP Cost [Valefor]
Rikku: Godhand - Double AP, Gillionaire
Kimahri : Spirit Lance - Double AP, Evade & Counter [Ixion]
Lulu: Onion Knight - Magic Booster, One MP Cost (it's possible to use 2 very high-powered Ultima for just 4 MP. Use Doublecast) [Shiva]
Tidus: Caladbolg - Evade & Counter, Magic Counter
Wakka: World Champion - Double AP, Evade & Counter [Ifrit]
Auron: Masamune - First Strike, Counter Attack [Yojimbo]
For customizing AP to your weapons:
Wings to Discovery*
Usage- 30 for Break HP Limit in armor
- 50 for Triple AP in weapon
Steal- N/A
S. Rare- N/A
Drop- Shinryu (Monster Arena)
D. Rare- N/A
Bribe- Marlboro (Calm Lands, Cavern of the Stolen Fayth)
- Great Marlboro (Inside Sin, Omega Ruins)
* 30 for geting 3 chests at chocobo racing
Door to Tomorrow*
Usage- 10 for Overdrive -> AP in weapon
Steal- Ultima Weapon (Omega Ruins)
S. Rare- N/A
Drop- Bomb King (Monster Arena)
D. Rare- N/A
Bribe- Mech Scouter (Calm Lands, Valley Bottom)
- Mech Leader (Mt. Gagazet)
* 99 for catching 6 areas
I can't remember where to get Double AP, but your best bet is to power up and fight creations at the Calm Lands Monster Arena, namely; Thu'ban, Neslug, Ultima Buster.
A tip for everyone's "legendary Armour" is to have Break HP Limit, Break MP Limit, Ribbon, and an extra slot spare (if you one for Rikku, you might want to add Master Thief ability).
I doubt writing the half the story would be appropriate right now, so I'm referring you to the relevent guides:
Ultimate Weapons - What you need to know
Monster Arena Bestiary - How to fight them
Bribe FAQ - Useful for getting the rarer items in the game. If you have equipment you don't use, sell them to raise cash.
If you select 'Customise' on the field menu, it will be possible to see the abilities which are open to you.
If you need to know anything else, just say.