Once you exit the bunker, you can handle the island in any way that you desire. If you want to play by the book (for now), then head west toward the nearby shrine.
The shrine turns out to be a church-shaped shed that spews toxins. There are three hostiles here, but they're not so much guarding the place as they are just hanging around. They can be dispatched with stealth takedowns alone, though the handgun works just as well.
One they're cleared, shoot up the big red tank to destroy the shrine.
Loot the site of throwables and ammo, then continue heading west along the road. This will bring you to the Forest Research Station.
Toss the place for ammo and supplies, but you'll especially want to check out the note in the main building, re: The Reaping. Reading this will mark the location of the other Shrine on your map.
Head there. The three hostiles here are more alert than at the last shrine, but you can get the drop on them by swimming over, and climbing up the seaside rocks. Swimming submerged greatly increases your stealth, and tangles of rope indicate particularly climbable ledges.
As before, destroy the shrine to complete your objective.