Easy mode:
Die three times on Normal difficulty.

Hard mode:
Beat the game on Normal difficulty.

Legendary mode:
Beat the game on Hard difficulty.

Pizzarmor:
Beat the game on any difficulty.

Rollerarmor:
Beat the game on Hard difficulty.

Guitarmor:
Beat the game on Normal difficulty - including level 30 of Necromandio Cave.

Hamburgarmor:
Beat the game on Hard difficulty - including level 30 of Necromandio Cave - with an "A" rank.

Udonarmor:
Beat the game on Legendary difficulty.

Sushiarmor:
Beat the game on Legendary difficulty - including level 30 of Necromandio Cave - with an "A" rank.

One World mode:
Beat the game on any difficulty.

Replay bonus:
Beat the game and save it. Now start a new game on any difficulty to begin with the same attack and defense stats and Mystic Stones from your previous attempt.

Thanks to Revolution readers Kris and Muath!