Civilization – Found religion in capital or border city

There are three methods of conversion: Organic, missionaries, and priests.

Organic conversion:

Organic conversion involves a system called "pressure". Any city with a dominant religion within 10^[1] tiles exerts 6^[2] Pressure. If a city has pressure from a religion, it will slowly convert to that religion. If it has pressure from multiple religions, it will try and balance citizens between them. A city with 12 pressure from christianity and 6 from shinto will gradually become 2/3 christian and 1/3 shinto.

Missionaries:

Missionaries are purchased with 200^[3] faith, and will have the same religion as the city they are produced in; if your empire is mostly Catholic, and you have a Protestant city, that one city will make Protestant missionaries. They can be used 2^[4] times to spread religion. This converts a number of citizens, based on several factors... how many citizens already have that religion, how large the city is, the strength of the missionary...

Inquisitors:

Inquisitors are purchased for 200^[3] faith, and they allow you to remove followers that are opposed to the inquisitor's religion from your city, once. (They do not work in opponents' cities.) This can make a minority religion into a majority religion. The inquisitor's religion is the same as the city it was produced in, regardless of your empire's religion. In general, inquisitors are not to be used for converting cities, though they can be used to quickly convert a conquered city to your religion if there is at least one follower of your religion in the city. Instead, inquisitors should be used to prevent the enemy from converting your cities. If one of your cities is experiencing high pressure from Hinduism, you can use an inquisitor to remove all traces of the infidels from your city. That buys you enough time to build support near that city, preventing the godless heretics from returning. In my play, I have been using inquisitors rarely when compared to missionaries. If you are not the dominant religion, you will not be using inquisitors to convert someone else.

Great Prophets:

Great prophets can be used like stronger missionaries and also function as inquisitiors, who have 4 uses and higher strength. They also do not suffer from attrition while in hostile territory, unlike missionaries

Notes:

[1]: There is an enhancement changing the reach to 13 tiles

[2]: Based on game speed. Slower games means lower base pressure values (4 for epic, 2 for marathon) There is an enhancement to increase the pressure values by 34% (and later 68%), rounded down.

[3]: Scales with game speed and era.There is an enhancement to reduce this faith cost by 20%.

[4]: There is an wonder (Great Mosque of Djenne) changing this to 3, for missionaries built in the wonder's city.

TL;DR:

To convert other cities, use just missionaries. Missionaries are offensive, Inquisitors are defensive.

A holy city can only be prevented from regaining a religion if the religion is destroyed. If it isn't, the city will be affected by internal pressure, usually 30. For the religion to be destroyed, there can not be a single city with that faith as its dominant religion. The holy city will continue to become holy again unless you do one of three things.

Option one: Station an inquisitor in the city. Problem solved.

Option two: Find every city with that faith as the dominant religion and change it.

Option three: Surround the holy city with enough cities of your faith so that the internal pressure is not enough to convert it back. This is probably 5, since internal pressure is usually 30.

PS: Don't ask me why internal pressure isn't always 30, I am not sure. It may regenerate over time after an inquisitor or missionary? I have seen it below 30 but never above.