I just recently started using storyprogression in my game, because I began to notice there were no longer any babies or toddlers in the game, asides from my own, and only a small handful of children left.
I like the ability to be able to sit back and relax without having to worry about whether the other sims are doing their....business, I guess you could call it.
But every few real life minutes I keep getting notifications of other sims getting promoted, gaining a skill, getting into relationships, moving in with one another and even funny ones like one of the sims being audited. I was wondering if there was a way to turn off the notifications without it affecting the other sims still doing their thing or if I need to keep the notifications on?? I don't want to mess with the settings without knowing what I'm doing, since this is the first time using storyprogression.

Choose SP > General Options > All Stories, and choose None or whatever subset you want rather than the default All. If you'd like to get more granular, each category under General Options, e.g. Options: Careers or Options: Romance, allows you to specify which sims generate notifications. This is independent of what SP is doing to advance these sims; it only changes whether you hear about it.

Keep in mind that it may take a sim-day or two, depending on the speed SP is currently running at, for already-created stories to clear the queue.