The quest is divided into several activities, and only the beginning is truly exploratory. If the squeaky aspect of this quest doesn't interest you that much, then maybe Bungie's Fourth Horseman video will get you through it:
Enter the safe
After accepting the quest, open the map and set your waypoint to the location of the quest. Even if the location is not far, it can be easy to get lost along the way.
There are several ways to get to the waypoint, but I prefer this shortcut: when you leave Zavala, you will see a few orange circles on the ground to your right. Go to the second circle furthest from Zavala and look over the railing. Descend onto the narrow bridge below and cross to the other side. Head towards the shaded structure and exit through the back.
Once you come out of the back of the shaded area, turn right and go down the stairs. Then take the first left on these stairs to descend to the lower level. After clearing the stairs, turn left and continue until you come to the open hallway on the right. After entering the hallway, enter Zavala's office on the left.
At this point, you can choose to go through those doors on the right to progress in the quest. Or, you can stay awhile and dance to Zavala's hidden workout mixtape. I highly recommend you check it out, along with the other Easter Eggs the room has to offer.
When you are done exploring the office, go through the doors, down the stairs, and enter a vent below the stairs you just came down. The rest of this part of the quest is pretty straightforward, as all you need to do is follow the vent paths. Wrong turns won't significantly deter you from arriving at the safe, as you'll be able to quickly turn around and correct your course.
Once you drop into the safe, you will realize that the Fourth Horseman is not there as expected, and you can grab the next quest step to continue.
Missing in action
When you return to Zavala empty-handed, he will give you the next quest step.
Fresh lead
Upon hearing of the missing Exotic Shotgun, Zavala tells you to visit Ana Bray. Head to the Hellas Basin on Mars to speak to him.
On the hunt
Anna thinks the Cabal has something to do with the absence of the weapon. The fastest way to complete this quest step is by participating in Heroic Public Events on Mars. You can make the Public Excavation Cabal events heroic by shooting down the Cabal ship. This quest step will progress by 20% for each Heroic Public Event completed, so you only need to complete five of them.
Back to Ana
Go back to talk to Ana, and she will tell you to collect more information from the Red Legion.
In the dark
The best place to complete this step is Firebase Hades in the EDZ. Chests grant 20% progress, so prioritize those who kill Psion.
Under our noses
Return to the tower to speak to Benedict 99-40, who will give you the next quest step.
Dirty work
Go back to the EDZ, preferably to Firebase Hades. Quickly complete this quest step by completing public aether replenishment events after making them heroic. To make them heroic, simply kill all the Servants before defeating the boss. Your eliminations will also count towards the progress of this stage.
Rule him
After completing the favor, return to the Tower to report to Benedict.
Lost and found
Head to the legendary Lost Sector, the Quarry, to finally grab your exotic Fourth Horseman shotgun after defeating the boss. Just note that the Lost Legendary Sectors are on a rotating schedule and will not be available every day.
To see the Fourth Horseman's stats and learn how to get the Exotic Catalyst, check out this guide.
