Becoming Suzerain of City-States in is a fairly straightforward process if you know what you are doing, but the process might be a little difficult to figure out without being somewhat familiar with the in-game menus and UI.
Becoming Suzerain of a City-State opens up unique gameplay options with the Independent Power and can give you strategic advantages throughout the Ages.
City-States are Age-specific, meaning they reset at the beginning of each Age. However, City-States can also be incorporated into their Suzerain's empire, or overthrown and razed, just like any other major Settlement would.
Independent Powers, which can be turned into City-States using Influence, are automatically loaded into each Age with unique bonuses based on specific categories that happen to sync up with the four Legacy Paths: Military, Gold, Science, and Culture.
When you first interact with a unit from an Independent Power, they will be hostile, neutral, or friendly. Any civilization in the campaign can improve its relationship with a given Independent Power, and it's first come, first served, meaning once someone claims an Independent Power, it's under their control for at least forty turns.