Culture also determines which tiles you get when yours and another civs' cities are close together. The stronger culture overwhelms the weaker culture.
Growth for your cities depends on how much

the city has in the tiles it is working. Each population point needs 2

to sustain itself; anything beyond that will contribute to the city's growth. For example, you have a city with population 4, working enough tiles to generate 11

, with no unhealthiness. It has a surplus of 3

, which will accumulate in its storage. Once the storage fills, the city will gain another population point.
Tax rate is the sliders in the upper-left. They convert

to

,

, and

. Anything left over is converted into

.
Trade routes are set automatically to maximize your total commerce. You need to have open borders to trade with another civ (otherwise your trade routes will be domestic). High-value trade routes go to large cities with good trade multipliers (e.g. Temple of Artemis, harbor, customs house).
Commerce is the

on the tiles your cities work. The main sources of

tend to be cottages and precious resources (gold, gems, silver). The city adds up the total

it gets from its tiles, and then divvies it up based on how you've set the sliders. Then it applies multiplier buildings. For example, if you have the sliders set to 0% except for an 80% research slider, and a city generating 20

, then that city would allocate 80% (16) of its commerce to :beakers: and 4 to

. If that city had a Library (+25%:beakers

then it would produce 20:beakers: and 4

each turn.
Capture workers by moving a unit that can attack (anything except workers, settlers, scouts, explorers, and of course oceangoing units) onto the worker's tile. You have to be at war to capture workers. The AI always destroys them instead; I don't know why.
You can only have 1 palace. If you build the palace in another city, then it disappears from your original capital. If you build enough courthouses, then you unlock the Forbidden Palace wonder, which acts like a second palace for maintenance calculations.