Having the same religion can get you quite a bonus, depending on the AI. Isabella will just love you for it, but beware not to switch religions on her or you'll be her next target. I think I saw +7 due to religion - not sure about it; definitely more than +4. I think also Monty likes you for your religion, but don't think for one second that that means he won't backstab you. He's the reason for: "your best friend is the person whose betrail shocks you the most".
Other ways to improve:
- open borders
- peace
- running their favorite civic when they also run it
- trading resources
- trading techs
- giving resources (even without them asking for them)
- giving techs (same as above)
- going in a war against the same civ
- having a defensive pact with them
- giving in to demands (techs, money, stop trading with worst enemy, going to war when they ask, civics switch)
I think that's it. Someone correct me if I'm wrong. Religion is by far the best modifier, but they all stack up. If you play it correctly you can get them to friendly. Don't try to do so with everybody, it's rarely possible since they will be mad for trading with their enemies, for having defensive pacts with others, etc, and it's hard to get everybody under the same religion.
That said, see this thread:
Lust for Gold. It's vanilla cIV, but it still applies. The goal was to start with Isabella and have the biggest shrine income by 1600AD. A couple of people, myself included, went the totally peaceful way, and converting everybody to the same religion to prevent any wars and any "worst enemy" cases was a very important part of my game. I made Mansa Musa switch 3 times (!) from Buddhism which he founded, and was going to do it any number of times needed to keep everyone happy. I gave him techs and money for it, didn't care, I was in front at techs anyway (my lead shrank later because of all the gifts) and I had no use for money anyway since I wasn't rushing anything nor upgrading units. I also gave techs and resources to both Vicky and Caesar to secure Defensive Pacts with them, since my military "power" was a little bit of a joke.