2 Civs I have often used are Gorgon and Pericles. Neither focus on religion, so you can have a religion while going for other things, like culture or domination. Basically, use their wild card policy. Rush to mysticism, and use the Great Prophet wild card for an extra 2 GP per turn. Still need to build shrine and do projects and race towards religion, but it helps alot of course.
On Immortal, I build a slinger (I do use a mod that gives free scout I just remember, so maybe ditch slinger or worker for a scout), then worker if it can be used. Else 2 settlers in a row. Then holy site, shrine and projects till I get a Great Prophet. That keeps the capital busy for a good while, and it is beneficial, to settle the other first 2 cities, with good initial production, so they can build some things themselves, mainly slingers and workers.
I do reroll to get good production in my capital, some 2 food to 2 production tiles, or better. Capital on a grass plain with no production, just feels completely bogged down to me.