To unlock Cody, Guy and Haggar for the arcade mode, you'll need to play through story mode and meet these characters there. Cody will be unlocked when Stiff and his goons take Cody away. Haggar is unlocked after you complete the part in story mode where you have to fight with Haggar against the punks. Finally Guy will be unlocked after you complete the part of the story where you need to escape from his burning dojo.
To unlock more levels in arcade mode, you need to also play through story mode. Pier district, Little Italy and Japantown are the levels you can unlock. To unlock the pier district, play through story mode up to the point where you meet 2-Ill. Save him and the level will unlock. Get to Little Italy in story mode and it will unlock in arcade mode. Finally Japantown is unlocked by completing the rat stomping game in Paco's tattoo parlor.
You can unlock a few music videos (featuring music not actually used in the game, madly) by completing certain story mode tasks. These music videos will then be unlocked in the "bonus" option at the main menu:
Fear Factory -Archetype: Unlocked when the Stiff has been killed.
Shadows Fall -What drives the Weak: Unlocked after you defeat Andore.
Shadows Fall -Inspiration on Demand: Unlocked after the second warehouse fight has been completed.
To unlock the original arcade version of Final Fight, play through story mode to the boss battle where Weasel is using a forklift truck against you. Defeat him and Fight Fight is yours to play. Sadly this version of it isn't very good, with a very choppy frame-rate.