Rate these INDUSTRIAL buildings in terms of importance

[stormwriter]

Lets say you're starting a custom game. Right off the bat you get these building choices. What is a good order to start building?

Walls
Barracks
Forge
Library
Observatory
Colosseum
Bank
Courthouse
Jail
(Harbor)

[quinalla]

It all depends on the situation, but I'll take a stab at it.

I am just assuming these cities have no culture generating buildings yet, that is why library is first even for the production city. If the production and food cities have growing culture, I would skip the library.
For a gold city:
Library
Harbor (if possible)
Bank
Observatory

For a production city:
Library
Forge
Barracks
Courthouse (assuming I have enough cities that it is worthwhile)

For a food city:
Granary
Library
Then Lighthouses/Harbors/Markets/Aqueducts etc. depending how unhealthy and unhappy the city is and what resources some of the buildings can take advantage of.

I know you didn't list some of the ones I mentioned in the food city, but I though it might be useful to say them anyway. In general, I find granaries very powerful in this game, so I particurally wanted to mention them. I usually build them in most cities and always in mid/late game cities.

Walls are situational for me, only build them if the city is in a precarious position. Colosseums only if the city needs them for happiness. I have yet to build a jail, but if you want to be able to have a long war, I imagine they help a lot.