The story in Gat Out of Hell does not progress like in Saints Row IV. Instead of it playing out with mission after mission, you need to fill a meter that appears on the menu’s screen, which is called Satan’s Wrath. As you do the side activities and diversions in the game, this meter will fill up and you’ll unlock more missions.
These missions are rather short and don’t have much to them, as it should only take around 4-5 hours to complete the main story in this game. Once you reach the thresholds shown as markers on the meter, you’ll usually see a cutscene and then unlock a new mission.