Red Mage Challenge:
Play the game with all chars using all equipment and all magic.
Of course you can freely choose which magics to actually learn, because of the lack of space.
I'd recommend Cure, Life, Heal, Safe, Shell, Change, Exit, Holy, Ultima, Fire, Ice, Bolt, Aero, Haste, Warp and Flare, though.

Mysidia Challenge:
Reach Mysidia, and buy equipment there until all chars are fully equipped, before heading for the Mithril Mines.

HP/MP Challenge:
Get 9999 HP and 999 MP before getting the Ring from Scott.