Is it better to focus a Great Prophet on opposing civilizations or your own at the beginning of the game?
There are two main cases I can think of:
- Say you are just starting out and you don't have a religion at all. Would it then be better to use it on your own cities primarily and then other civilizations after?
- Say most of your cities are already converted. Same question: would it be more beneficial to continue converting your own cities or to move on elsewhere?
There are a bunch of factors, so the answer depends heavily on what's going on in your religion and your Civ.
Religious pressure will emanate from cities where your religion is the majority religion. This pressure can convert cities within 10 tiles, but it's kind of a complex topic. It's probably safe to assume that, in the absence of other religions, your religion will spread to your cities eventually.
Thus, if you've got no other religions nearby, and you don't mind waiting, send your Great Prophet to other Civs to spread the good word. If you're in a rush to convert your own cities, and/or other religions are putting pressure on your cities, keep it local.
Also your beliefs matter. There are a lot of beliefs that only count foreign cities (Peace Lover, Pilgrimage, etc) so if you've got some of these in your religion you will benefit more from spreading it to other civiilzations.
On the other hand, some beliefs benefit the city rather than the religion's owner. Spreading your religion to another civ's cities might inadvertently give them a small boost, which you might not want.
It should also be noted that Great Prophets will remove opposing religions when they convert a city. So if you've got a lot of other religions nearby that are infecting your citizens, you can "cleanse" a city and potentially get it back under your control with a Great Prophet.
To summarize, there's no one right answer to this question. Consider what benefits your religion has, how important they are to your current cities and other civ's cities, and what the state of the spread of your religion is. The combination of these factors will tell you if it's better to stay close to home or send your prophet on a journey.