There is a "passive" spread mechanism in the game, where a city with a religion slowly "pressures" a neighboring city to convert to its religion, but it is quite slow. Far more effective are missionaries (cheaper) and apostles (more expensive, but with certain promotions, much more powerful). You can only buy missionaries and apostles with faith (you can't purchase them for gold or build them, like other units) and only in a city that (a) has a religion and (b) for missionaries, has a Shrine, and (c) for apostles, has a Temple. You can get two free apostles if you build Mahabodhi Temple (a world wonder) or, if you are playing as the Kongo (who cannot found a religion), if you build a Mbanza or a Theater Square in one of your cities that has an imported religion, you get one free apostle of that religion.
To use a missionary or apostle to spread your religion, walk the missionary or apostle to a target city and, once they are standing in a tile adjacent to that city's center tile, choose the spread religion action button. That hits the city with a burst of religious pressure, converting some number of citizens to your religion. You may have to "zap" a city multiple times to convert it to your religion (having a majority of its citizens following your religion), depending on its size and whether there are other embedded religions in that city already.
Civ VI also has a concept of religious (or theological) combat, where religious units of different religions do "combat" to determine who gets to sway the hearts and minds of citizens of neighboring cities. Apostles can both attack and defend, while missionaries can only defend. A third type of religious unit, the inquisitor, can both attack and defend as well, but it cannot spread religion to new cities - it is used to defend your religion in your own cities (it gets a significant theological combat bonus when in your own territory) and can be used to "purge" other religions from your cities. When a religious unit is defeated in combat, two things happen: (1) the victorious unit emanates a surge of pressure for its religion that affects all nearby cities and (2) the defeated unit emits a surge of "negative" pressure, reducing the influence of its religion in the same nearby cities.
