Kingdom Hearts/Championship Cups

Phil Cup
The Phil Cup is unlocked by Sealing the Traverse Town Keyhole.

At around level 26 it should be an easy task to do every mode

Rewards

 * Initial = Gravity (or Gravira)
 * Solo run = Mythril Shard (or Combo Plus in 1.5 Final Mix)
 * Time Trial = AP Up (or Tech Boost in 1.5 Final Mix)

Pegasus Cup
The Pegasus Cup is unlocked by Completing the Monstro World.

Rewards

 * Initial = Strike Raid
 * Solo run = Mythril (or Orichalcum in 1.5 Final Mix)
 * Time Trial = Defence Up (or Dark Matter in 1.5 Final Mix)

Hercules Cup
The Hercules Cup is unlocked by Sealing the Halloween Town and Neverland Keyholes.

At level 56 it should be an easy task to complete every mode.

Rewards

 * Defeat Cloud: Metal Chocobo.
 * Initial: Herc's Shield, Yellow Trinity, and Olympia.
 * Solo run: Orichalcum (or Critical Plus in 1.5 Final Mix)
 * Time Trial: Power Up. (or Gravity Break in 1.5 Final Mix)

If you get to Hercules in the time trial and it doesn't look doable, you can restart the round and the timer will freeze giving you all the time you need to beat him. (KH1 only)

Hades Cup
The Hades Cup is unlocked by Sealing the Hollow Bastion Keyhole.

Rewards

 * Defeat Yuffie = Genji Shield
 * Defeat Behemoth = Blizzaga
 * Defeat Cerberus = Thundaga
 * Defeat Leon & Cloud = Lionheart
 * Defeat Hades = Ansem's Report 8/Graviga
 * Defeat Rock Titan = Trinity Limit
 * Solo run = Save the Queen
 * Time Trial = Save the King

Gold Match
After completing the Hades Cup, a special match will be unlocked against the Ice Titan. Note that this is only available in the North American, European, and Final Mix versions of the game. Firstly, it's best to have Curaga and Firaga in your shortcut for easy access to these moves - they're likely the ones you'll need most. Use Dodge Roll to evade his attacks and jump onto the stands to get a better view of his head. Lock-on to his head and attack it with Firaga. For defeating him, you get 5000 EXP and the Diamond Dust keyblade exclusively in Final Mix.

Platinum Match
After beating all other cups and locking the Hollow Bastion keyhole, the Platinum Match will become available. Like the Gold Match, it is only available in the North American, European, and Final Mix versions of the game. The Platinum Match pits Sora in a battle against the antagonist of Final Fantasy VII, Sephiroth. Give Sora the Strike Raid, Second Chance, and MP Rage abilities, as those can be essential to winning the battle. Use Dodge Roll as much as possible to evade Sephiroth's attacks. To win this battle, it's best to be above Level 55. Winning the battle will get you 18000 EXP and, in Final Mix exclusively, the One-Winged Angel keyblade and Ansem's Report 12.