How to get the story mode data back

Yes, the Social Club account is device independent, as long as each device uses the same login details for the account you play the game with.

It's obviously just the Social Club that is shared though, not achievements and game progress.

Main Missions are denoted by either M or F on your map (and another letter later in the game, I don't want to spoil you). By doing these missions you will inevitably advance in the story. Sometimes they take some time to pop up due to daytime or just because you haven't fulfilled a prerequisite such as having a phone conversation. But these prerequisites will be quite apparent so don't worry about them. What I would suggest is to switch characters once in a while and explore a bit. The mission should start eventually.

I could provide more help if you could tell me what story mission you did last. Please refer to the GTA Wikia for a list of main missions.