if i have a settler factory, every other city will build a worker as soon as it is possible since i dont want to disrupt my settler production for making workers. Depending on food/shields that means it either starts with a worker, or makes one after 1 warrior or curragh.
After that, it depends on the situation.
During the expansion phase, i will normally keep millitairy to a minimum to feel a bit safe. That means very little millitairy. I will first try and do some needed investments for the future. Often i start a granary after the worker.
If after that, i still don't think i need to focus on millitairy, a marketplace is next (if i don't need an aqua yet).
Sometimes however, millitairy power is needed and i will build a barracks and units. I always try to build units only AFTER i have a barracks in the city.
Building the barracks can vary from right after founding the city to after the marketplace. This obviously depends on "Foreign Politics". (I am a probably a warmongerer in that in my mind, conquest is the only real victory, but i do use my brains and choose my timing for wars carefully)
Cultural buildings have lowest priority, i occasionally build one in a high flip risk city or in a city that needs border expansion for a resource or something.
Many cities never get any culture at all.
I am used to play deity/sid, there you cannot get close to the AI's culture anyway, so no need to build any for that. For happiness, i just as well use the lux slider as paying upkeep for a temple so there it is no priority either. Researching is useless on those difficulties, so no libraries either. Only later on when cathedrals are available, i sometimes start to build some things for happiness.
barrack's are only build in cities close to the capital who have good production. They will build almost solely militairy units from then. Half corrupt cities do not get a barracks, and will build artillery units. Total or near total corrupt cities dont get any improvements at all. They will build workers, artillery units or be set on wealth.