The important thing is as TSJ said, pick a couple civs, who like each other, and get on their good side, and then discipline yourself to tell everyone else to pretty much go spit.
You cannot please everyone, and in fact unless the entire world runs either the same religion or no religion, the task is impossible. The most common mistake I would make is trying to trade with everyone, and ending up having worst enemy red modifiers for everyone, which completely negated what I was trying to do.
In a perfect world, the couple of civs you pick to be buddy buddy with, will not be civs you plan to go to war with later, especially if you are doing a lot of resource per turn trading with them. Nothing worse then finally going to war with a civ, and then realizing that the instant you go to war you lose 20 gpt of trade, plus 2 or 3 each happy and health cap trades. That can hurt worse then war weariness.
-Sinc