To unlock harder difficulties for subsequent playthroughs of the game, try beating each stage once through in their entirety. Then, when you play the stages again, you'll have the option of playing on harder difficulties.
To unlock an area called Home, where you can communicate with characters from Story Mode, complete Story Mode once through in its entirety.
To unlock a New Game+, beat the game once through and start a new game using your old save file.
To unlock New Game++, beat New Game+ in is entirety.
To unlock River Belle, complete the first story mode. After completing Mt. Vaal for the second time, you can go to River Belle.