Beat the game then complete the following tasks to unlock the corresponding flag.

Unlock Arlong's Pirate Flag:
Defeat Arlong a second time around

Unlock Black Cat Pirate Flag:
Defeat kuro the second time around

Unlock Buggy's Pirate Flag:
Defeat Buggy the second time around

Unlock Kreig's Armada:
Defeat Don Kreig the second time around

Unlock Navy Flag:
Defeat Axe Hand Morgan or Chaser the second time around

Unlock Cook Pirate Flag:
Get Sanji and talk to Chef Zeff in 4-1

Unlock Alvida Pirate Flag:
Defeat Alvida in 1-1 then find her in 6-1

Unlock Veggie Pirate Flag:
It's located behind the Veggie Pirates in 3-1 (you need Ussop to get past them).

Unlock Straw Hat Pirates Flag:
Collect ALL the flags but the red hair pirates flag

Unlock Red Hair Pirates Flag:
Beat boss battle mode in ALL difficulties


Unlockables
Complete the following tasks to unlock the corresponding item.

Unlock Wado:
Complete level 1-2

Unlock Zolo's Stomach Band:
Complete level 1-3

Unlock Axe-hand Morgan Statue:
Complete level 1-3

Unlock Chouchou's dogfood:
Complete level 2-2

Unlock Map of the grand line:
Complete level 2-3

Unlock Special Buggy Ball:
Complete level 2-3

Unlock Chakram:
Complete level 3-2

Unlock Cat's Claw:
Complete level 3-3

Unlock Usopp's Sling shot:
Complete level 3-3

Unlock Mackerel Head One:
Complete level 4-1

Unlock Free Food:
Complete level 4-2

Unlock Gas Mask:
Complete level 4-3

Unlock Lollipop:
Complete level 4-3

Unlock Hatchan's Octopus Trap:
Complete level 5-2

Unlock Master of the Near Sea:
Complete level 5-2

Unlock Tooth Gum:
Complete level 5-3

Unlock Elephant Tuna On level 6-1:
Have Sanji talk to a woman

Unlock Shigure:
Complete level 6-2

Unlock Billow-A-Bike:
Complete level 6-3


Unlockable Modes

Unlock Boss Battle Mode:
Collect ALL the Coins

Unlock Mini-Game Mode:
Successfully complete the game once