While you'll be able to unlock most of the soundtrack by finding the right pickups in the various levels, there are a few songs that you can't get without beating the game first. After you've defeated Yin-Yarn, you'll unlock the Dream Land, Yin-Yarn, and Staff Credit Theme music.
Dream Land Music
Beat the game.
Staff Credits Music
Beat the game.
Yin-Yarn's Music
Beat the game.
Staff Credits
The last cutscene you unlock also includes the game's staff credits. You'll need to beat the game first, and you can rewatch them any time in the Flicks menu.