From the Apolyton University:

City States can be your best friends.
rah's guide to leveraging one of the biggest additions to CIV IV.

I know some of you have seen city states and did some of the math. Hmmm, I need to get to 60 influence points for them to be my ally. So I can give them a lot of money and for a few turns I'll get a little benefit. I'm here today to tell you that your math is slightly flawed because you don't understand just how much return you're getting for your money. The strategy that I'll be outlining is more grand in scope but it's simple. You will ally with every single CS in the game. Now why would you want to do that? BECAUSE IT"S FUN. And you'll find that if you start any game using this strategy, every victory condition will be available. This is for Single Player only.


The overall strategy is that you're going to go vertical on the tech path. You will race to the Medieval Era so you can unlock the patronage tree in your social policies. So your goal is to go vertical to either get Theology or Civil Service. You can get to Theology in fewer turns but doing Civil Service allows you to build Chichen Itza which makes golden ages last longer, and some people really like to make sure they get this.
But first let's choose a civ. The concept will work for almost any civ but there are 5 that can make it easier. Below are a list of those civs and how they can help.

THE EGYPTIANS
Since a couple of early wonders are really helpful, their ability to build them quicker can help. And on top of that, since culture and happiness are so important to the game, their UB is one of the best. The Tomb provides 2 happy and 2 culture WITH NO MAINT. cost. You will have better things to do with your money. Their UU doesn't require horses and is early so you can use it to protect yourself early in the game, when your focus on this strat will make you a little vulnerable. This is the civ I prefer to use because you'll spend more time building wonders that don't require maint over building actual building that do. With marble and the social policy Aristocracy you can crank wonders.

THE GREEKS
The special power of your City State influence degrades slower and recovers faster will save you a lot of money and let you ally more city states earlier. Their two UUs are both available early (again when you'll be most vulnerable) and may be the some of the best UUs available. But I'm not here to discuss that.

THE SIAMESE
Their special power also applies to City States. Food and culture increase. Also more bang for your buck.

THE BABYLONIANS
Their special feature gives you a Great Scientist that you can use to get to CIVIL SERVICE or THEOLOGY much more quickly. (and if you want to try other Medieval techs, you have more flexibility.)

THE FRENCH
Their special power gets you culture faster and gives you benefit for the early part of the game which will help you work the entire patronage policy group.

BEGINNING THE GAME. (one approach)
OK, you've selected a civ so let's cruise. While you'll figure out many ways to get a Medieval tech you can try targeting Stonehedge and the Great Library. At any decent level you can't screw around here. It's best to build Stonehedge first then the GL and complete both before turn 50. The babs special power makes them research writing which encourages an early build. Or if you're not the Egyptians (which is why I like them) them wonder bonus usually nets them an early wonder also.
So every turn counts. Since there are no bad ruins you have to aggressively search for them, (and wonders) so you really can't not build a scout first. If you got good land you'll be on a river with river wheat. Any hex that provides production works but it's a hex that can be improved the second you kick out that worker (that you will build when the scout is done). River Deer also works but it will be later before you can do it. Using a one 1 1 1 square gives you the production, but since science is based on pop you really like to keep growing as quickly as possible. If you can open a ruin that adds pop, you can get away with worse land.
So you're working that river wheat or equivalent and 5 turns, in your scout is out cruising and 14 turns after that your working is farming that wheat. And your starting warrior should be done searching and back home to protect you worker.
Since you want Stonehedge and the GL your first three techs will be POTTERY, CALENDAR, WRITING
Your worker will be usually completed prior to finishing on Calendar so you actually can make a choice here. You'll have 5-7 turns or so depending on what else has happened. Start on that monument and switch when Calendar is done. Or another warrior, (it never hurts)
OK, Stonehedge is done and you're now banking some serious culture. Now here's a big thing. By now, you've probably amassed enough culture points to open your first social policy and you'll be asked to select one. DON"T, just keep saving up those points until you've gotten to the medieval era. Just ignore it.
You should have writing and researching the tech that will lead you to the medieval era. (I usually do philosophy) Check your timing so you don't finish GL before you research the tech that leaps you. I.E. make sure you're be done with philosophy so you can choose theology as your free tech. Ok, now you've reached your goal , choose your medieval tech and advance to that era. NOW you can use your culture points to open Patronage and probably have enough left to select both Aesthetics and Philanthropy. Now the stage is set. Yes you will want to finish off the entire patronage policy before starting on any others. I'll usually not settle a second city until I've completed it so I can get there faster.
Now it's time to start to buy city states. (as long as you've got the first two patronage levels completed and have 500 dollars. ) You should be close to 500 at this point just from turn gold, ruins, and barbs.
Choose a City State.
You can do this however you want but what I usually do is look at the ones close. They're easier to protect and not as likely to be attacked. And if they ask you to do something, you're closer (if that even matters) I like doing a Maritime one first since I want the food to jumpstart my city growth. Second I like a Military one since I would prefer never to have to build more than one or two more units. You like the military one close so the gifted troops don't have so far to go. Third I'll either do a second Maritime or maybe a cultural one. If you have choices, look to see what their resources are. They've rigged it so they all of them will have pretty much the same resources, maybe 3, 4, or 5 to choose from. Try to spread which one get to help your happiness. Keep in mind if you can't sell resources that you get from City states but if one is providing you with one you already have, you can sell yours and still have theirs even though it only listed (1) in your trading screen. Once you have another 500 dollars, rinse and repeat UNTIL you own ALL the city states.

WHY WHOULD I WANT TO OWN ALL OF THEM.
There are a lot of reasons. And here is some of the fun.
Let's assume you are allied with some maritime cities. Your cap will have so much food it will quickly grow and allow you to work production and gold squares to fuel your civ. And your other cities get less but the benefit is big with each additional one. It's like granaries without having to pay any maint.
Due to the social policy you're getting 33% of their science production. When you add that to the boost you get to yours due the pop explosion, it's easier to stay ahead of any AIs technology wise.
Due to the social policy you're getting great people gifted. Getting an engineer is always good for building wonders. Or a merchant gives you more money and influence, and you can figure out what to do with the others.
Free units, If you look at the math, getting a unit every 10 or so turns it's cheaper to buy off military cities then buying them in your cities. If you're not thrilled with the unit you get, feel free to gift it back to the City state. Yeah, you only get a few points of influence but later on you'll want your City states armed. They're capable of doing that themselves but it's cheaper if they're picking up the troop maint.
The culture boost from cultural CSs could be better but you'll be amazed at how fast it adds up. Every social policy gives you an advantage so the more the merrier. If you'll going for a cultural victory they're really helpful.
The strategic resources are really good. I've had games where I don't have very many strategic resources nearby. Now I don't really bother with those much. The can supply all you need. Yeah you have to pay attention to make sure you're not totally relying one that will be targeted but in general if you're allied to all of them you'll look up and see you have more resources than you'll ever need. REALLY.
The biggest fun is when you declare war. RELEASE THE HOUNDS. I've seen CSs conquer civs. You don't even take the happiness hit when they're out razing cities. It also keeps those civs busy so you can take out the juicy targets. If you're out teching the AI, so will your CSs in terms of combat units. Sometimes I'll declare war on someone on the other side of the world and all my CSs will nibble till they're offer you almost everything for peace and you never even moved a unit into their territory. And it seems that when I own them all, civs are reluctant to declare on you, and if they do they find themselves busy quite quickly. It's an auto diplo win 10 turns after the UN is built. If you're trying for a different victory condition, you may want to turn off the diplo win since it can happen quite early. It's even more hilarious when an AI builds it for you.
Take a few choice AI cities and it's not hard to have access to EVERY happy resource which keeps your huge cities handled happiness wise, and due to yet another social policy the resources they're giving you count as additional happiness.
So while each advantage seems small and not worth the investment, you must consider the overall picture. (and it's pretty)

OK, THIS SOUNDS GREAT BUT HOW THE HECK TO I PAY FOR IT?
Besides careful management of your empire to generate money, and the extra you get during a golden age, the best way is to sell all your resource to other AIs. (YES I SAID ALL, but I'll usually keep marble if I'll be building a wonder in the next 30 turns.) Early in the game, since you delay the second city, you really don't need excess happiness. I will sometimes wait for that first golden age since it's near when you're ready to start buying. Later in the game you won't have to sell all of them, but you'll be hooked and continue till you run out of buyers. And selling gives you an idea on how much the other AI's like you. The new diplomacy is under the hood so I can't guarantee that it works this way BUT, it seems if they like you, you can get 300 for a resource, if they're not thrilled with you, 250, and if they really don't like you 100. So you have to manage your diplomacy a little. You'll be amazed at how much you can generate if you're selling ALL your resources. But money is only one way to keep them happy.

INFLUENCE WITHOUT GOLD.
Again there is strength in numbers. If you own all the CSs, almost anything that happens will please someone. One CS gives you a great scientist and 2 others had asked you to create one so your influence with them goes up. Almost any wonder you build makes someone else happy. Discover a map wonder and someone else is happy. Take out a barb camp and someone probably asked you to do it. At first, the CSs continual whining to have you do things for them annoyed the heck out of me. Now I don't even pay attention. Whatever I do will make someone happy. Juicy if you can kill a barb and return a worker to a CS. 30 free influence points.

THE BIGGEST INFLUENCE WITHOUT GOLD.
Liberate them. Sometimes this requires some maneuvering. But if you liberate one they grant you so much influence that you very very very rarely ever have to give them another dime. If you try to own all of the CSs the AI do notice and will try to take a few (they'll do it anyway) So park a couple of units near by a CS you might think at risk. If the AI captures it, the next turn while the city defense is at 50%, retake the city and liberate it. FRIENDS FOR LIFE. If they've asked you for help when they were attacked, (and they always do) and you don't think the AI can take them, Declare on the AI and destroy 3 units and it's not as good as liberating them but it's pretty good. I've declared on an AI just to tease one to take one a CS just so I could liberate it.
If you're doing all these thing, you won't have to part with as much gold as you think. Now if you think you have better uses for that gold, what is it. CSs provide things that you would have to build and pay maint on. Don't need to build/buy troops. You can spend your time building the free buildings that don't require maint. And a few science and culture buildings in cities that are focusing and maybe a few happy buildings if you're doing the conquest thing. When I kept my civ small, I've run happiness at +80 and higher.

IN SUMMARY.
This isn't the bible, just remember the concepts and execute however you think best. BUT remember, City States are you friends, and you can never have enough friends.

rah.