Find the ‘Profiles’ folder of GTA V

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.

For me, it works this way:

You have 2 files for 1 savegame, for example: SGTA50000 SGTA50000.bak

You need both of them in the folder (C:\Users\USERNAME\Documents\Rockstar Games\GTA V\Profiles\SC_NUMBER). Try deleting all older savegames.

Restart the game and click on "Story Mode", the save will load automatically.

If this doesn't work:

1. Backup the "Rockstar Games" folder (C:\Users\USERNAME\Documents\Rockstar Games\)
2. Delete the whole Rockstar Games folder (C:\Users\USERNAME\Documents\Rockstar Games\) => This is safe, I also did that. But your game settings will be gone if you don't back it up.
3. Restart the game, the folder will be re-created new with all the needed data.
4. Put your newest savegame-files into the folder (both files): C:\Users\USERNAME\Documents\Rockstar Games\GTA V\Profiles\SC_NUMBER\
5. Restart and try again.