I play expansion heavy in broad strokes at Immortal level -- I want X amount of cities so I have X amount of unit pumps, X research and gold builders, so much of the map controlled in my culture, etc. So I tend to have dim views of lots of buildings unless they can provide a wide benefit, which generally leads to me evaluating the cost in hammers/opportunity of a building compared to whether it improves my empire's standing.
A big factor in viewing a building favorably is cost reduction from traits as well, so I'm sorta of a mind with georgejorge and I do agree with what that poster says regarding the early stages --after you get Alpha/Currency there is little consideration for most buildings for a good while unless you are doing something specific (this coastal spot needs a lighthouse) or are warring earlier than later, though my cut-off point is even earlier...I don't consider building anything higher tech than a forge or courthouse until it's time for the Factory-driven hammer economy, build wealth and GPP generation is just too powerful before that stage of the game.
A run down of the stuff I actually will build might look look something like:
Granary - important, every city eventually needs one though your +food surplus and whipping frequency modulate how early you really need one. Getting a granary as your very first build slavishly in each city is actually ignoring a lot of what might be a better option at the time (unit for fog busting? Monument for necessary border pop? need another worker while Cap works on next expansion? etc etc), especially if it doesn't really help that city (lots of food surplus) or that city isn't being whipped hardly at all --paradoxially, this can be because the city is weak in food surplus like a +2fpt city growing on cottages
Monument - always worth it for a CHA leader in the first 50% of the game, no question --they can be 1 pop whipped for no net happy loss, and provide a border pop and net +1 happy 10 turns after that. Otherwise not great and situational to the city site. Sometimes 100% necessary for border pops (offshore fish, etc) where waiting on an AI religion or Library (sometimes Writing can be delayed) is not feasible. In any case, waiting on Code of Laws or Music to pop your borders is a WAY worse option, and I view Mysticism tech in the same way as Wheel -- it is much better to get it sooner when you can afford to research it than later when you have need of it and can't!
Lighthouse - always worth it in any city with a fishing resource, I don't care what those hardcore players who are way better than me say

+1 food an a tile you ALWAYS going to work is huge boon -- see settling on a Sugar resource, or the Supermarket. Also, as I tend to econ crunch before Currency in most of my games, coast tiles are preferable for short term digging out of the hole to non-river cottages (especially when I am stretched on worker turns!), and a Lighthouse is necessary for this. ORG or FIN leaders have even more bias towards them.
Library - very situational if you are not CRE. Libraries are expensive and take a good amount of productive resources (3 pop or chop + 2pop!) in the very early phase, while the greatest benefit they give (allowing you get to GS early) is dependent on the food surplus of their city site. They go in strong locations that can afford to whip them before the Alphabet > Build Research cutoff. If CRE, they go in pretty much any spot up until about Currency as they just blanket increase your tech rate, but they still aren't worth giving up Build Wealth turns for. CRE civs can also use them for border wars quite easily, which may even be necessary if you get unlucky with a pushy CRE neighbor.
Barracks - if you are AGG these are amazing for handling your barb problems with Cover/Shock/Woodie 2 warriors super early. Otherwise I tend to build them later during the phase when I am whipping war infrastructure all at once -- Forge + Courthouse + Barracks in each city. If I am Cuir attacking, I'll already be in Nationhood too, which makes them even more valuable on a mass scale for their +2 happy.
Stable - specifically for cuirs attacks, I'll whip these in all mature cities (small pop or weak island spots skip it) so the cuirs come out with 2 promos -- 3 promo cuirs are extremely effective and 5XP gets them halfway there. They are also really good for generating a lot of overflow into either the 1st cuir, or to complete the barracks before it.
Forge - this is first thing I whip going into war mode in every city that can actually grow. Try to get it down to a 3 pop whip, though 4 is okay if you get full (~30) oveflow. These make the subsequent infra whips much more manageable, grant you some extra happy (pretty easy to have at least one of the resources through expansion/trade by this point) and create the opportunity to chain-whip even expensive things like Cuirs with just a few turns waiting on regrow/getting over the 2pop whip threshold. After war they improve your Build Wealth yield!
Courthouse - especially important in the mid-game war phase, though typcially too expensive if not ORG or Gilgamesh to bother with before then. You cannot Build Wealth while producing units, and large empires gained through conquest are EXPENSIVE especially since the AI sucks at its developing land before you take it. The courthouse reduction not only stands in for the typically similar amount of Wealth gold per turn, but reduces your static costs across the empire even as you whip the entire thing down or gain more cities. Having courthouses already in place when the warring ends makes it considerably easier to recover too, and are good first whips when captured cities come out of revolt.
Factory and Coal Plant - no brainer once it's time for the hammer economy. In lots of games I just Build Wealth until Assembly Line after the war phase, whether that's done with Axes,HAs, Elepult or Cuirs. In games where I can peacefully(ish) expand into most of my surrounding land I may even just build Wealth from Currency until AL.
Health buildings (Harbor, Aqueduct, Grocer, Supermarket, Public Transportation) - these become necessary as you implement Factories + Coal Plants and you have the additional unhealthy from power and coal/oil. Without them, your cities start to hit a wall on growth as they start to get larger. Fortunately, once you have Forge + Factory + Coal Plants, these builds become pretty easy.
After that, the only other buildings I may build are Laboratories for space part building, assuming I go Space and not Modern warring.