Here is what I do:
1st Policy: Tradition Opener (every single time, regardless of who or what map you play.)
There is absolutely nothing more useful then the +3 culture and getting more land for your capital early on. Even on Diety, if the AI actually wins a cultural victory, you've been a complete fool. There has never been a game of Civ 5 I have played where my culture isn't 3-10 times more then everyone else, to where it's impossible to reach with tourism.
2nd Policy: Liberty Opener or Legalism (unless you are playing small. In which case fill out tradition)
3rd Policy: Liberty Opener, Legalism, or Honor Opener (If there are Raging Barbarians)
Then Finish out Liberty or Tradition
Open Piety when you are ready to start building Temples. Religion is best when you have a lot of cities. Stay busy buying religious buildings.
Get mandate of heaven if you have pilgrimage and are very interested in spreading religion.
Otherwise, just wait until Industrial era, because increasing your faith and making buildings cheaper doesn't do you any good.
You'll just wind up with a bunch of prophets you don't really want.
Get the extra faith AFTER Renaisance, and then fill out the rest later on. Save your writers and an artist for the worlds fair.
But consider carefully which great people you really want. Unless you're playing Venice, you should only be making Scientists.
If you have a lot of happiness and wonders built in several cities, filling out Ascetics is much better, especially if done before Worlds fair.
Because what great people do you really want to buy? Engineers are useless in the late game. Merchants are nice for Venice, but in all liklihood gold generation won't be a problem. Admirals are useless. Generals are okay, if you want to steal land, but you should be earning those through war. An Artist or Two is okay. Maybe buy one or two musicians if you are going for cultural victory, but if doing that you'd want to fill out Ascetics anyway.
The bottomline is: Writers and Scientists. That's all you really want to buy with the Reformation, and the first one or two of others. If just writers, Ascetics is better, BUT opening piety, getting the extra faith generation, and Theocracy is good, and should be done anyway.
Keep in mind that to get the Reformation, you have to get Mandate of Heaven, Religious Tolerance, and Reformation, that's three policies you usually don't want, just to buy writers, that you will use to get more policies. IF you actually win the Reformation and the AI doesn't get it first. It does little more then pay for itself. It's better to go 2 piety and fill out Ascetics instead.
Full early Piety is not useful. You have to wait until Industrial to buy them anyway.
It's pretty much just whether or not you get Mandate of Heaven and that's if you are trying to dominate religion and maximize your founder belief.
But if there are other strong religious civs, like the Celts, you probably should'nt bother.
I once won a cultural victory on Emperor during the Renaisance era because of Piety and Religion:
First off, I got a lot of faith runes. Save your first prophet while rune hunting and you will keep getting them. When you have enough faith for the second prophet, then found the religion, buy a building, and you're halfway to the next building.
I got Mosques and Pagoda's that game. My neighbor, Egypt, made the second religion, he did early piety, got Cathedrals, and Monasteries, and Sacred Sites reformation.
I was playing Venice that game, btw. I captured one of Egypt's missionaries, and started using missionaries and inquisitors to spread his religion, and then I wound up with a full empire of his religion, with all 4 buildings in every city, each generating +8 tourism.
I think I entered the Renaisance with like +50 tourism which is absolutely sick at that level of the game. The AI just couldn't generate enough culture to out pace, so I wound up winning a CV I think in the late renaisance or early industrial.
Piety is better suited for the AI. Let THEM buy the Prophets, Missionaries, Enhance the Religion, and get the Reformation, and then steal it all from them!