This is just what I use currently as my rough picks/method. I probably also play quite different maps/settings than most. My typical game is/probably will be for all time 4 player, duel/small, pangea/1 island each, modified quick, no tech trade, culture flip after conquest. Results may be quite different for more sparesely populated maps, larger maps, and epic/normal speed games.
In the government ones:
Heriditary Rule +1 happiness per military unit in city
Leave use default until you are having happiness problems, are changing a different civic anyway, or use
Universal Suffrage +1 production from towns
I love it, I have lots of towns. If you do not have lots of towns, stick with default/hereditary or use
Representation +3 research per specialist, happy people in largest cities
I've used this all of once or twice, I never seem to have enough specilists, and Suffrage isn't too far off usually.
Legal Civics:
Vassalage +2 unit built xp, some free units
Great for when you are building military units in peacetime(if you run out or feel the need for more), and during wartime (since you are obviously going to bebuilding more)
Bureacracy +50% production and gold incom in capital
More for the production, great for small maps, useful if I get trapped to two cities, gives a nice boost.
Free Speech +2 town gold, +100% culture
As I said, I love towns. I also happen to love culture. I switch to this when I am done hevily building military. In wartime it may be beneficial to switch to Vassalage again.
Labor
Slavery spend pop for rushing
Much the same as hereditary rule as far as when to get. Maybe after you have both, change both civics at once, but I still usually hold off until one or the other is needed. Tack it onto a different civic revolution.
Emancipation
By the time I get this, just about everything is a town anyway, but might be useful to recover from pillages, also no upkeep.
Economic Civics
Free Market +1 trade routes
Better than default
State Property +1 food to windmill/watermill, no distance upkeep
The other thing I have a lot of is windmills.
Religious
(couple notes, I don't really care about founding religions, but what religions I do get, I spread in my country, monastaries give nice culture and science, and temples make people happy. I generally keep no state religion to avoid wars early(it happens before my economy is set if I take one.))
Organized Religion +25% building production, missionaries without monastaries
If I do happen to have a state religion because all of my powerful neighbors have it, i'll take this.
Free Religion +10% science, +1 happy in each city per religion in it(i think)
I run no state religion usually anyway, same upkeep as default, with extra science and happy people, yay. After I get in the general tech area, I try to nab this. Also gives my other favorite civic, Free Speech.
My typical advanced civics are:
Universal Suffrage, Free Speech/Vassalage(war or peace unit building spree), Emancipation/Slavery(emergency war), State Property, Free Religion
