I'm currently playing my first successful game on immortal. The most important thing to keep the AI from attacking you is to have a strong enough military that you aren't an easy target. This means that you might have to forego wonders and other early buildings to catch up to the AIs starting units. You should also focus on scouting early, so you can meet as many city-states as possible, find some ruins, and kill some barbarians, all of which boost your economy. In my experience, it also means that you're going to have to attack someone early on in order to catch up to everyone else's growth. A few good puppet cities (preferably with one of those wonders you skipped) will put you back on par with the other AIs.
Some other tips regarding relations I've learned
- Try to get a pact of cooperation early on, while there's still plenty of space between you and the other civ. They aren't time limited, so it can help protect you for a long time.
- I've been turning down pact of secrecy offers, they limit my ability to be friends with as many people as possible.
- I've also been turning down open borders, this just allows the AI to scout me and see that he likely has a military advantage.
- I didn't settle many cities (only one in fact, the rest of my cities were captured). Eventually the AI settled near me, and this did cause our relations to sour, but at least it put it off.
- I've never given money to other civs, but I have occasionally given them free resources if they ask for them nicely.
- Research agreements are good, but be careful that you don't help one AI get too big of an advantage. I try to make agreements with the civs that have a lower score.
At the higher difficulties, I've found that the only way to win is to play defensively. Siege units, at least one in each city, probably flanked with another siege unit (or two on a border with an aggressive enemy) and some front line infantry is really the only protection.
My second city is almost always founded in an area with a lot of production bonuses (hills, iron, stone/marble, horses/cows/sheep, etc) and focuses completely on producing military for the entire game. If you grow large enough, founding an additional city of this type can be useful, but if you've got around half a dozen cities or less, one is generally sufficient. Have this city crank out units and only build military buildings there (ideally by rushing them, if you've got the cash). Most of the rest of the empire is money- and happiness- focused, although they might churn out disposable units if war is looking inevitable or going poorly.
When it comes to science, pick techs that get you closer to:
- Money production buildings, like the Bank
- Better siege units (trebuchet, cannon, and especially artillery)
- Better front line infantry (musketmen, infantry, mech. inf., etc)
The AI will become increasingly aggressive towards you if they feel your military power is less than theirs. However, they're terrible military strategists. They will send their entire army right to your cities, with no regards to how that leaves their defenses or how positioning effects their chances at survival. At the higher levels of difficulty, this can mean an almost constant trickle of units.
However, they tend to fail to prepare for the bonuses you gain for fighting on your own turf - take these whenever possible, and then turtle heavily when they declare war. After a while, you'll have an army of extremely powerful units (ie, 2-shot per turn siege units with extra combat range, etc) and great generals, which you can use to push the tide back. Plan on cranking out front line infantry units, as they're somewhat disposable. Put a large priority on preserving your high-level siege units, though.
Likewise, if you see a weak neighbor, go after their stuff. The AI gets big bonuses to production, so their cities are going to be better equipped than ones you could found yourself. Declare war, turtle for a few turns while they bumble about ineffectually at your borders, and then steamroll.
Just remember not to completely abandon your own defenses...
Best Answer
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.