Pantheons I think can be either used for your long term strategy (ie, Science Victory, Cultural Victory) or can be used for a short term boost (over the long haul, Pantheons aren't that powerful, but in the short term some of them can be very powerful in the right situation).
Long term:
Wide
Ancestor Worship: +1 Culture for every Shrine in your civilization
God of Craftsmen: +1 Production in cities with population of 3+
Messenger of the Gods: +2 Science in cities with a Trade Route
Goddess of Love: +1 Happiness from cities with population of 6+
Tall
Fertility Rites: 10% faster Growth Rate
Goddess of Protection: +30% increase in city Ranged Combat Strength
Monument to the Gods: +15% Production of Ancient/Classical Wonders
Mother Goddess: (increased food production)
Religious Settlements: +15% faster border growth
Terrain
Dance of the Aurora: +1 Faith from Tundra tiles without Forest
Desert Folklore: +1 Faith from Desert tiles
God of the Open Sky: +1 Culture from Pastures
God of the Sea: +1 Production from Fishing Boats
Goddess of Festivals: +1 Culture and +1 Faith for each Wine and Incense
Goddess of the Hunt: +1 Food from Hunting Camps
One with Nature: +2 Faith from Natural Wonders
Oral Tradition: +1 Culture from Plantations
Religious Idols: +1 Culture and +1 Faith for each Gold and Silver
Sacred Path: +1 Culture from Jungle tiles
Sacred Waters: +1 Happiness from cities on rivers
Stone Circles: +2 Faith from Quarries
Warfare
Faith Healers: +30 HP healed per turn if adjacent to a friendly city
God of War: Gain Faith if you win a battle within 4 tiles if your city
As you can see, most Pantheons are based on your terrain. Notice how all the terrain bonuses are either culture or faith (except for camps). Personally, I think it is best to pick a bonus that will give you the most help in the first 100 turns or so, not to line up with your overall victory (your Religion should do that job instead). I feel like faith bonuses are particularly strong in the early game (unless you've got really good terrain you still need cultural infrastructure and CS's to really make a dent in SP costs) because you only need about 500 faith to get yourself off to the races faith-wise, and a really good bonus (like say, Desert Folklore in an oasis town can easily trump Stonehenge). That might be different at higher difficulties where it is harder/impossible to be the first with a religion or even found a religion. It doesn't take much to really milk a terrain pantheon (especially folklore, sacred path or auroras). Even 3-4 culture or faith in the first 50 turns is going to double your normal output at that point.
When would you pick a non-terrain bonus? First of all, If you don't really have a good terrain setup, you might pick a long-haul bonus. The production bonuses are situationally strong. Craftsmen is essentially Republic 2.0 and you'll probably be getting both in a wide strategy. The first few hammers in a city are always really powerful (the second hammer is essentially doubling your production rate
).
Messenger, Ancestors, and Love Goddess are good for a wide strategy, but are more beneficial later on so you might pick one that lines up with your victory type.
If you've got marble, and are going for a tall cultural victory (or just have a really early wonder you want to make sure you snag) then Monument looks really good. I just finished a game with Egypt and took this and OMG Wonders! Again, probably weaker on high difficulties since wonders are hard to get.
Settlements to me look like a tall bonus, unless you are going wide, are never going to open tradition, and want to discourage the AI from settling in between your cities. I don't see them being very useful - a wide empire either conquers or places cities to take territory, and a tall empire takes Tradition pretty reliably. Food is always good for tall strategies, but might cause happiness issues in a wide empire. Protection is for turtling up in your cities and teching/culturing your way to victory.
TLDR:
Priority list go!
1. Pick a terrain-based faith bonus if you have the matching territory and think you have a shot at founding one of the first religions (difficulty dependent).
2. Pick a terrain-based culture bonus if you can get say, 4+ cpt from it in the first 50 or so turns.
3. Pick a production bonus that fits your overall empire management (Wide = Craftsmen, Tall/Wonders = Monuments).
4. Pick something that fits your long-term strategy (example: Messenger for wide science victories, Goddess of love for domination or puppeting strategies).
Obviously, for certain strategies, especially civ-specific ones, you might pick differently, but this is what I've been operating from on a general basis.
Relgious beliefs:
Founder beliefs give benefits (mostly happiness/gold) for spreading your religion. Follower beliefs give benefits (mostly faith/culture/happiness) to your own cities for your religion. Enhancer beliefs are for helping you spread your religion or to help with warfare.
Founder:
Ceremonial Burial is probably the strongest (compare 1 happiness to 1 cpt or 1 fpt) especially for a wide/puppet strategy. Gold just supports everything in this game, especially helpful for Tall strategies where you won't have as many trading posts, and Tithes and Church property are great. I feel like faith at this point is less strong since you are no longer worried about getting a religion first. World Church and Pilgrimage seem weak. 12 cities converted is a lot harder than just allying with a cultural CS.
Follower:
Pick what fits your overall strategy. I always like to pick one that lets me build a building (Pagodas are awesome in general, Cathedrals are nice for the extra specialist slot for cultural victories), and then pick one that gives a general benefit. That way you are still benefiting from faith once you're done spreading/enhancing your religion but you haven't hit the industrial era yet. Hammers are generally scarce in this game so Religious Community seems pretty strong (goes well with Craftsmen or, situationally, God of the Sea pantheons).
(Does anyone know if Religious Art provides the benefit only in the city where you build Hermitage or in every city with the religion? I haven't actually taken it yet).
Enhancer:
Religious Texts is really strong for spreading your religion. Just War is nice for conquering cities on your own continent. Defender is good for turtling strategies, or for defending toeholds in enemy territory. Messiah seems to combine the benefits of Holy Order and Missionary Zeal. Missionary Zeal might be really powerful with the Mosque of Djenne (I haven't tried it yet). Reliquary seems undertuned, and again your infrastructure should take care of most of your faith needs.
, + 1%
per follower.
from Tithe, the best AI (Rome) spread its to 5. In last game the same Rome spread to 20 earlier than I even found these cities.
