There are a lot of good answers here, but there are some things that haven't been mentioned.
Religion plays a big part in the life of those who are a part of it, if not controls most of the actions of one's life.
In most religions, you are going to have some sort of prayer, and dedicate time to that prayer.
Some religions only pray on certain days (i.e., Sunday).
Some religions pray once per day.
Some religions pray multiple times per day.
Some religions have set time for prayer, that could be associated with of the above.
You could create a character that would have to dedicated a certain amount of time to prayer per day. This enables you to create story elements such as "if you don't pray at 5pm every day, for at least 30 mins, your chances of turning into a demon increase by 35%" or something along the lines of your character continually gets closer to becoming a demon, and you have to pray to negate that effect, at a certain point.... or something else to make the story interesting.
Since you specifically asked
I'm looking for ways to make their religion get them into adventures.
You could use the above in order to search for items that could aid in your story.
For example, if you find the "Book of Ages" you can decrease time spend on prayer per day, or decrease time from when last prayed, to when you need to pray again, which would enable you to spend more time doing other things in your world.
Religions can have events where you need to collect specific items, which might be in distant lands, and prepare them in certain ways according to tradition.
This can enable you to go on adventures in 2 ways, while creating more story elements.
Find the items for the festival in distant lands, or at least acquiring these items will require an adventure of some sort (Slay the Dragon and acquire it's tail for the brew of tonight's feast)!
Learning to cook, or create items for the festival could bring you to interesting places in your world.
This adds story elements, because your characters could build experience in certain things such as "Crafting, "Alchemy," or some other skill-set you have enabled in your game.
It also adds the story element of these items being useful in your adventure later on. Maybe you learned how to cook a special healing food that is an ancient item that appears in this special festival once every 500 years, and is passed down in a special way which you had to spend a lot of time, money, etc to acquire.
There are many possibilities for "Religion" to be a part of your story.
You could have a religion of demonic summoners who use their own life as summons.
or a religion of people who are able to create golems, i.e., put life into objects (possibly also using their own life, or some sort of energy).
In reality, you need to look at what a religion is. You said that appearances is something you noticed i.e., Monk, but there's so much more than that. A religion is the bond between you and something else, most notably a God, or some other Deity, but there are idol worshipers who might worship a structure, or other "idol."
Religions have shown that if you are "religious enough", that you can gain something. This is the important thing to note.
You pray to your god enough and he will grant you the ability to summon the demons above, or create golems, heal people, gain the God's ability, etc.