The main story of Shelter 69 is told through the Campaign missions and their related H-scenes. Additionally, side stories are frequently included as part of regularly occurring Events. Lastly, there are Champion Stories, detailing the backstory, relationships and character arcs for the Champions that join your shelter. These are told through a combination of campaign missions, Champion missions, Trust missions, Flirts, H-scenes, and events.
If you're curious about the order of champion stories and events relative to the main story, you can see an overall timeline for the Shelter 69 story here.
The lead writer for Shelter 69 occasionally provides additional world building details and stories on discord. As much as possible, that information can be stored here.