There isn't much story to follow. Survival and running around is what the game is mostly about. Or walking around, as running costs stamina. And mostly about survival of your cubs: I am not sure if it is even possible to get the mother lynx killed. She does not absolutely need to eat, it only helps her run longer, and lethal things like drowning just make her pass out for a bit and wake up with starving cubs. Survival is mostly about hunting, as starvation is the main cause of death for your cubs: in the vanilla game your cubs can die from starvation or wolves, and wolves are rare (I think they are only in the tundra, so don't go there with small cubs). Moutains DLC adds foxes and eagles, which may eat your cubs, or failing that, get eaten by them. Once your cubs are strong enough to leave the den you can wander anywhere you like as long as you can find food.