Many new players, it seems, do not practice diplomacy in an organized, coherent way but in a "spur of the moment" haphazard way. For example, an AI request will come up which asks you to "stop trade with someone" or "give tribute/help", "declare war", etc and I think many beginners just decide at the moment what "feels" best with no long term planning. But this isn't the best or most interesting approach.
What you should do, upon founding your empire and exploring the land is to see who and where the AI Civs are located and how powerful they are, then decide who is going to be your your long-term ally. Then using the F4 screen, figure out who another Civ that the potential ally really likes (and presumably that third civ likes back) that you also would want as an ally. Then you will attempt to (as much as possible) get relations between you and these two Civs as high as possible, forsaking all others when necessary. The key to remember is that you must ally with two Civs that themselves are extremely friendly to another thus avoiding the "trading with enemy" penalty and also the awkward situation where each ally asks you to war on the other! This usually but not necessarily requires you to have the same official religion as these other two Civs.
And that means that when they ask for reasonable tribute, you give it. That means when they ask you to stop trade with those outside the triangle, you do it. That means when they ask you to join in war, you do it if it is safe to do so (and it is a huge bonus to do it as it prevents the -1 you failed to help us in war but rather gives +1to4 for mutual struggle). Realize though that when you do that, that other Civ will be a lifelong enemy most likely but that okay. Of course, it means that relations with some of those Civs outside of the triangle will probably be irreplably harmed but it is much better to have strong friends and strong haters than to have everyone cautious or annoyed at you.
Now it is important to avoid having "backstabbing" or "rogue" or particularly warlike nations in the triangle. Although all AIs will backstab if you are too weak or present too much of a target (so in all cases having a credible military is a must), some do it much more than others. Montezuma and Isabella come to mind in most cases. Avoiding warlike civs is also good because you might otherwise be forced into too many wars and being a pariah nation yourself.
But it is important to also realize that you can shift or change the triangle as the situation unfolds. Unless you're playing with very few Civs, if you're playing with 8 or more, its unlikely that you have great relations within the triangle but all others hating you. Say you have a triangle with A and B but also have good relations with C and A also have good relations with C. You may then decide to have C in the triangle and drop B. So in this way, you can shift the triangle from time to time. But by focusing on two at a time, you will find the game proceeds much better than trying to equally please and equally displease everyone.
What you should do, upon founding your empire and exploring the land is to see who and where the AI Civs are located and how powerful they are, then decide who is going to be your your long-term ally. Then using the F4 screen, figure out who another Civ that the potential ally really likes (and presumably that third civ likes back) that you also would want as an ally. Then you will attempt to (as much as possible) get relations between you and these two Civs as high as possible, forsaking all others when necessary. The key to remember is that you must ally with two Civs that themselves are extremely friendly to another thus avoiding the "trading with enemy" penalty and also the awkward situation where each ally asks you to war on the other! This usually but not necessarily requires you to have the same official religion as these other two Civs.
And that means that when they ask for reasonable tribute, you give it. That means when they ask you to stop trade with those outside the triangle, you do it. That means when they ask you to join in war, you do it if it is safe to do so (and it is a huge bonus to do it as it prevents the -1 you failed to help us in war but rather gives +1to4 for mutual struggle). Realize though that when you do that, that other Civ will be a lifelong enemy most likely but that okay. Of course, it means that relations with some of those Civs outside of the triangle will probably be irreplably harmed but it is much better to have strong friends and strong haters than to have everyone cautious or annoyed at you.
Now it is important to avoid having "backstabbing" or "rogue" or particularly warlike nations in the triangle. Although all AIs will backstab if you are too weak or present too much of a target (so in all cases having a credible military is a must), some do it much more than others. Montezuma and Isabella come to mind in most cases. Avoiding warlike civs is also good because you might otherwise be forced into too many wars and being a pariah nation yourself.
But it is important to also realize that you can shift or change the triangle as the situation unfolds. Unless you're playing with very few Civs, if you're playing with 8 or more, its unlikely that you have great relations within the triangle but all others hating you. Say you have a triangle with A and B but also have good relations with C and A also have good relations with C. You may then decide to have C in the triangle and drop B. So in this way, you can shift the triangle from time to time. But by focusing on two at a time, you will find the game proceeds much better than trying to equally please and equally displease everyone.
It took quite a bit of diplomatic maneuvering, trading and going to war when he asked but has paid off many many times over. 
