Man Eat Dog World is the first side mission in Far Cry: New Dawn. You can get it from the NPC found near the first captured outpost. He will tell you about a dog captured by bandits. The NPC will also mark the location for you.
This location is a slaughterhouse. You don't have to look for a way inside. Before you save the dog, you will face only two bandits. Both of them are in the same spot - in the room where animals are killed. You aren't able to avoid fighting with them. Eliminate the weaker bandit first. The other enemy is well-armored and carries a shotgun that deals massive damage. Don't let this enemy get closer to you. The best tactic here is to fall back and hide behind covers.
Save Timber the dog. The animal's cage is open. Interact with the dog - it will point you where you should go next. Go to the opposite side of the room with the bandits - you will find an entrance to another room. Go inside and then follow the dog.
You will find a climbable wall. Follow Timber. You now have to move on the cages' rooftops - keep doing that until you reach the place where the dead dog is. Master Cage Key lies next to the animal's corpse. This key allows you to open every cage in this building. Use it to collect resources left here. Go back to the place where you have rescued the dog - let its companions free. This completes the mission. You also get a new gun for hire - Timber.