The Story menu is where you can access all the available Date and XXX modes, with any of the girls you've unlocked so far:

Selecting a Story will take you to its respective Date story roadmap.

The Puzzle shaped missions are Date Modes, and the Heart shaped ones are XXX modes.  

The Text Bubble mission is a conversation which can often also contain decision making ( line picking) in order to score the best outcome.

The other Icons are items that can be collected, like XP, Sparks, Orbs, Outfits, Stickers and Animations.