On Deity the rules for picking beliefs changes quite a bit. The AI will grab many things before the player can and you can only reasonable target getting the 5th religion. The following is good for sub-Deity play. Some of it carries over to Deity but not all of it.
Pantheon:
When picking a Pantheon your top priority is almost always to generate faith. This greatly narrows down what beliefs you have to consider. You want to pick which ever belief will get you more faith over the next 50-100 turns.
Good faith generating Pantheons in order from strongest to average are:
- Desert Folklore: (+1 faith from desert tiles) - head and shoulders the strongest belief if you started on a desert (or maybe even near one).
- Stone Circles/Tears of God: (+2 faith from quarries/+2 faith from Gems and Pearls) - Grab if you're in a geographical cluster of the resource.
- Religious Idols/Goddess of Festivals: (+1 faith +1 culture from Silver and Gold/+1 faith +1 culture from Wine and Incense) - Grab if you're in a geographical cluster of the resource.
- Earth Mother: (+1 faith from Salt, Copper, Iron) - Grab if you're in a geographical cluster of the luxury resource.
- One with Nature: (+4 faith from natural wonders) - usually inefficient unless you can get 2 natural wonders, but you can take it with only one if you're desperate (or Spain).
- Dance of the Aurora: (+1 faith from Tundra tiles) - Much much weaker than Desert Folklore and rarely worth grabbing. Usually the only tundra tiles worth working are deer and luxury resources (Desert Folklore can get floodplains and oasis).
If you can't make use of one of those than reasonable fall back pantheons are:
- God King: (+stuff in your capital) - Get stuff for no extra effort.
- Messenger Gods: (+2 science per city connection) - Science is a valuable yield and city connections are easy to make.
- God of Craftsmen: (+1 prod in cities of 3 pop or more) - If messenger is gone.
A few other pantheons can be reasonable fall backs in certain situations.
Founder
There are only a few good founder beliefs and the default grab should usually be Tithes. If you need Happiness than Ceremonial Burial can be worth it. After those two it gets into niche strategies. For example, Interfaith Dialogue can be combined with cheap missionaries to become an effective Science generator.
Follower
Follower beliefs can be clumped into a few categories:
- Faith bought buildings: (mosques, pagodas, cathedrals) - They are good but require a strong faith engine to make use of and probably mean less of a faith stockpile to buy great people with during the industrial era.
- Happiness sources: (+1 hap from shrines/+2 hap from temples) - If you need happiness these beliefs are pretty much the best way available in the game.
- Situation beliefs: This includes Religious Centers (+1% prod per follower, max 15) and Swords to Plowshares if you don't need happiness. Monasteries (+faith and +culture from Wine and Incense) if you're in a geographical cluster of Wine and Incense.
- Everything else: The dregs. Rarely worth taking.
Enhancers
It is common to just default into the passive spreaders, Itinerant and Religious Texts. Holy Order (cheaper Missionaries and Inquisitors) is part of the Interfaith Dialogue combo but end up consuming your faith which means less great people in Industrial and probably no faith-bought buildings.
Messiah and Just War have parts in some niche strategies but everything else is the dregs.