Try to get Holy Site+Shrine up on your capital ASAP.
I try to get a religion every game, and early enough that I can get some good picks.
I actually failed at getting Dance of the Aurora in a game playing as Russia, with a nice Tundra start, with Dyes in that wooded tundra (read: +1 faith if worked, +2 faith if worked as Russia, from turn 1). Turns out that Sumeria was sandwiched between two holy CSs, so they were getting +4 faith per turn from just about turn 1.
The first point, one I didn't see mentioned, is to not forget that Faith and Great Person Points are two totally different things now. Its easy to slip back into the Civ V thinking. For me, anyways. So, for example, if you used the +1 faith/gold card to found a pantheon, make sure to switch it out for +1 production to build that shrine and holy site.
People often chop wood for Stonehenge... well, you can chop for the shrine/temple, too. And then chop for a setter to get a military city up and running before going back to Holy Site Project. If you have to chop to clear a site, I would switch to settler, chop, then switch back to holy site.
Usually won't have to run the Holy Site Project more than once, if at all. I find that civs will build the holy site, but not the shrine. Keep an eye on the Great Person wall of shame, and scroll over the points/turn the other civs are getting. Do so frequently.
Remember the cost keeps jumping up as new era's are reached. If I recall, it goes 180 to 360, right? Doubles? So you might think you are behind and will miss it, then hitting the new era buys you breathing room.