How to save story progress?

I have no idea, but here is how you can try to fix this problem:

1. Thanks to our fellow member Elena for this one (How do I turn autosave on?)

"Its on the Settings tab, actually. (for me at least) Pause>Settings>Saving>Autosave

In PC: Pause>Settings>Saving And Startup>Autosave"

2. Thanks to our fellow member Dom for this one (How can I save my progress?)

NOTE: Only works when not in mission.

Text about the autosave which may help:

The auto save feature is pretty neat, as it'll save after every completed mission, mission replay, random encounter, activity, in short: anything of relevance. The auto save feature won't save after you improve a skill, however, and if you're in flight school or in the shooting range, you'll have to leave those before the auto save feature does its job.

However, the quicksave feature (accessible from the cellphone) is pretty neat too. Try it out, and you'll never want to go back to your safehouse to save, never. Well, maybe you'll occasionally want to return to change clothes, watch TV, fetch whatever vehicle is in your safehouse garage, or to time travel 6 ingame hours into the future. The beauty about the quicksave feature, is that it'll save your current position, the vehicle you're currently driving (but not its damage, which is kinda nice) and is just perfect to abuse the stock market. Just quicksave in front of the assassination missions, start the mission, jolt down the stock you'll want to invest in, load the previous quicksave, then invest in those stocks as everyone (as you won't be able to switch while on assassination mission).

Basically, auto save does a good job, but quicksave does an awesome one. Use it well and do it often. From our fellow user Nolonar (Do I need to save my game manually?)

Hope this works :)