Missions from the Priest can give you a big reputation boost for that nation or clear any hostilities, but you won't get any real reward, except the loot dropped from enemies. It takes about 5 priest missions to jump from "Neutral" reputation to "Admiration" - a big leap compared to the Governor's quests.
Guard the church
The priest tells you that some strange characters have been coming recently to church at night and he's afraid that they might be robbers. He asks you to guard it for 3 nights - you must be in church from 0:00 to 6:00 in order to complete the mission. Sleeping in the tavern will not work, so pass the time by other means (exiting into the world map is the fastest way, though you will need to time it correctly so you don't overshot midnight. Saving before exiting into the world map will prove useful in this regard.) until the time is nearly midnight. Now, enter the church before 0:00 and wait. If nothing happens that night, at around 0:00 to 0:05 you will simply get a black screen with the words: "Morning has crawled her way out of bed". That means you'll have to repeat everything and be back in church before midnight again. Apparently, even after 3 nights, you still won't be able to turn in the quest so you would have to indefinitely guard the church until the robbers appear. If no black screen appears and robbers appear (usually there are 3 of them), after a short chat you will need to kill them and then rest till morning, go back to the church, and then speak to the priest about watchman duties to receive your reward.
The Priest of the local church will sometimes give your character a quest to clear the evil spirits from a nearby cave (or dungeon). To complete the quest, you will need to enter the correct cave/dungeon.
If you have entered the correct cave, your logbook should update when you spawn the skeletons by running into the glowing white light, and you won't be able to leave the cave so bring plenty of potions and antidotes before entering the cave and killing all the skeletons. Once you have slain all of the skeletons, the door should unlock. Return to the priest for your reward.
A note on the caves: Sometimes, it is difficult to find the correct cave. For instance, if you received the quest in Santa Catalina which has a cave just outside its walls in the Jungle. However, the quest cave is actually a different cave, south of San Juan Del Norte beach, towards Porto Bello. This is why I recommend looking at a map of the specific city/island where you got the quest, to make sure you aren't missing any caves, or just watch for the logbook update when you walk into the glowing orb and spawn the skeletons.
Based on original from Skodarap's CoAS Site (used with permission). Original is here .