You have two general city governor options: production automation, and population management (these are the two buttons that are right above the six small buttons in the governor window).
You should generally leave the left one alone (production automation), unless you like if the governor decides what to build in the city
The right button, population management, can be useful however. Without other settings, the governor will try to have moderate growth, production and commerce. This is usually ok, until your city grows to a medium size. At that point the governor will attempt to create specialists, and here you may have to intervene: if you see that the governor puts up a specialist that you don't want, turn the gov off, re-arrange the specialist as you want it, and turn the gov on again. Often he will respect your choices, however sometimes he may put the specialist back as before (can happen if the city location dictates it, for example in a production poor environment the gov WILL make engineers or priests no matter what you tell him).
The six small buttons below the automation buttons do nothing on their own, they only work if the population management is active. Be careful to use them, as they will often remove any specialists you had carefully set up. Best experiment with them for yourself
On your question of culture, I think the best way would be to set the governor to "emphasize commerce" (on of the six buttons). More commerce = more culture. "Emphasize great people" is NOT good for that, because the governor (in my experience) rarely sets up artist specialists. The other (and often best) option would be to turn the governor off here, and set up artists manually.
Hope this helps a bit
