I found that the Patronage social policy tree was quite beneficial for keeping my city-states happy. Their relations never drop below 20, I get more bang for my buck with cash donations, and their favor drains less quickly. With all those bonuses, one donation of 500 gold brings them immediatelyWays to "allied" status, and keeps them happy for quiteKeep/Improve Favor with a few turns.city-state:
The Patronage Social Policy Tree - this tree has can help improve the amount of influence gained with cash donations, slow the decrease of influence over time, and improve the "default" influence level on all city-states. These policies make a huge difference.
Playing as Alexander the Great of Greece gets you the "Hellenic League" bonus: "City-State influence degrades half as slowly as normal, and it recovers at twice the speed as for other civilizations." Also, Alexander has an undocumented bonus that trespassing in the territory of another city-state does not cause you to lose influence with them.
Pay attention to the disposition of the city-states you ally with. "Friendly" nations will lose influence slower, and "Hostile" nations lose influence faster. Favor Friendly allies to save money.
Do Quests offered by the city-state.
Liberate a conquered city-state. Liberating a conquered city-state will greatly increase your favor with them. City-State must have been conquered by an opposing nation first. Liberated City-States will also Always vote for you for Diplomatic Victory.
Free stolen workers - if another player has stolen a worker from a city-state, capturing the worker will give you an option to give it back to the city-state it originated from. I only had this happen once, but I think it gave me more influence than gifting my own units to them, if I'm recalling correctly. (Please correct me in the comments if this is wrong)
Gift Units to the city-state.
Gift Money to the City-State.
If you don't want to invest in Patronage, you're going to have to do their quests or cough up the money. Having good cash flow is critical for expanding your influence on city-states. I find that connecting trade routes to my capital is a great way(Edited to boost my income in the early game, before I havesummarize a large population and markets/banks/etc. Get those roads out and get your empire connected. As long as you don't spread your cities so far apart that you kill yourself on road maintenance, it's easy money. If you have cities onnumber of good points made by other continents, remember to build a harbor to connect them to your trade routes.posters)