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I would say the time to bring up the backstories of each character is when it feels right and feels worthwhile. That said, it may make sense to jot down some notes on what those backstories are no...
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#3: Attribution notice added
Source: https://writers.stackexchange.com/a/41628 License name: CC BY-SA 3.0 License URL: https://creativecommons.org/licenses/by-sa/3.0/
#2: Initial revision
I would say the time to bring up the backstories of each character is when it feels right and feels worthwhile. That said, it may make sense to jot down some notes on what those backstories are now for later reference. Your suggestion to save the apocalypse for later in the story makes sense to me. The people best suited to rebuild a world are probably not the same ones who were best suited to stopping the everything from coming apart more. Or maybe the apocalypse ended because some people just managed to survive until it was over. In either of these cases, the people who would be the focus of the story may very well not be dwelling on what happened, but thinking ahead to the future or about how to survive for today. My personal recommendation for your first draft is to get as much of the detail down as you can so you don't forget, and leave refinement to later drafts. As they say, your first draft will always suck. It's more important to have a first draft that helps you get to the end with a story you like and your fans will love than it is to minimize the number of drafts. Disclaimer: I've never published any literature. I've never finished anything to the point of being suitable to send to a publisher, or even an editor. My goal is to advise you against making my mistakes. Remember you can always change your path, so long as you remember enough of the world of the story you want to tell.