The diplomacy game in Civ (=RFC/W) is more difficult than you would think. Basically, what I've come to understand is that there are approximately three ways to go. I will describe these three general strategies; perhaps there are others as well.
1. The Partisan
This is probably the most basic and obvious strategy, but it's not an easy one to take. Basically you aim to make good friends of as many neighbors as possible (theoretically they don't have to be your neighbors, but you probably want to keep your neighbors happy more than you care about the farther-away civs.) The most important thing is to pick a small cluster of thick-and-thin friends, and then make friends of their friends, enemies of their enemies. Once you go to war with a friend of your friends you may be surprised how quickly it all falls apart. As a pacifist this strategy can work very well. I seldom play the pacifist. One of the best way to please your group of friends and keep you all together is religion. Pick your friends, pick the religion they use, and work hard at getting them all to have that religion in all of their cities so they never want to leave it. A general tip that is always true is to sign Open Borders with anyone you want to be friends with ASAP, as it improves your diplomacy with them, especially over time. It also gives you money.
2. The Isolator
This is probably the easiest way to go, but it's kind of risky. Your general strategy is not to rely on other civs. Trade with everybody equally, don't pick friends and enemies - except for the purpose of wars - and never have a favorite. Be completely opportunistic. Friends and enemies will come and go, nobody will ever really love you, and rarely will anybody really hate you. You don't actually depend on anybody else so you can really do whatever you want, to an extent. A large army will help keep you safe.
3. The Kingpin
This is basically a variant of the Isolator strategy only usable when you're third or second place, or sometimes first place. So you play one of the two other strategies until you're one of the top three. Once you're there, the idea is to play powermonger. You favor the weak civs, and the enemies of your enemies, and any weak civ in a position from which some more power could harm a strong civ. The most important part of this strategy is not to ever bow down to anyone's demands. Your general trend should be to level the playing field by weakening the other top five civs and strengthening the bottom five civs. This can draw a lot of fire away from you, and when you're high up there on the score chart drawing fire away from yourself is vital for your survival (until such a point when you're so far ahead of the rest that you simply can't lose anymore.) Remember that even when you own all of Europe and the Americas, you might have a hard time holding that land against a dogpile of all the remaining civs in Asia and Africa. As the Kingpin, you should use wars purely as a tool of diplomacy, though major campaigns of conquest don't actually hurt as long as you stay mindful of diplomacy. A good way to hurt a friend of an enemy is to declare war on them and buy every living weak civ in the world into the war with technology. This will cause a world war to break out in which your enemies will be busy harming themselves. And whoever loses the war is disposable to you anyhow. Whoever ends up on top can be your next enemy. If a neighbor happens to fall, that's an opportunity to spread your control to another city...or ten.
In fact, war should be treated as an extension of diplomacy in all strategies. This simple rule is what makes choosing your friends as the Partisan so hard. You have to think ahead about who you may want to conquer, and keep them on the enemy side of the fence. Preferably, their neighbors should be on your side of the fence, unless you want to take them in as well. For the Isolationist all of this is somewhat less important because the whole strategy is to be inconsistent and free.