I feel you need a pretty good starting position for your capital. I build scout, slinger, worker, settler. Sometimes you can squeeze a trader in too, go for a gold city state if you can, or the second city to help production mainly. I do the three starting techs then go for Holy Site. As soon as I can build it my cap begins, then builds shrine then do faith boosts till I get a religion. Beeline for the civic that allows you a great prophet wildcard.
I will usually buy a worker for the second city, and it has to play mini-capital until the religion is secured. Basically building worker/military to support.
Keep an eye on the great people, the 2 first religions will go without a chance. The last 3 prophets will be a race, take note how many Great Prophet points the rivals are getting pr turn, and keep boosting as long as you have to.
After that your cap will need granary, windmill and campus. Basically kicking it into action. You'll want 2 more settlers as luxuries support 4 cities. I am not sure it's always a good idea to build them in the cap. As the city lose 1 pop, and you need that pop for third district (merchant) to get things going.
If you can buy a settler, then don't do it in the capital.