Worlds (Servers) are separate from the cities.

There's three major city-states you can start in - Ul'dah, Limsa Lominsa, and Gridania. Which you start in is based on which class you picked to start as. When you progress to level 15 in the main quest, you will be sent as an envoy to the other city states, and from then on can travel freely between them.

Each server is an exact copy of the game environment. The same places, events, content, etc., exists on every single server; the only difference are the people.